Original: L EARNING C UT S ELECTION FOR M IXED -I NTEGER L INEAR P ROGRAMMING VIA H IERARCHICAL S EQUENCE M ODEL
Revised: L EARNING C UT S ELECTION FOR M IXED -I NTEGER L INEAR P ROGRAMMING VIA H IERARCHICAL S EQUENCE M ODEL

Original: Anonymous authors
Revised: Anonymous authors

Original: Paper under double-blind review
Revised: Paper under double-blind review

Original: A BSTRACT
Revised: A BSTRACT

Original: Cutting planes (cuts) are important for solving mixed-integer linear programs (MILPs), which formulate a wide range of important real-world applications.
Revised: Cutting planes (cuts) are important for solving mixed-integer linear programs (MILPs), which formulate a wide range of important real-world applications.

Original: Cut selection—which aims to select a proper subset of the candidate cuts to improve the efficiency of solving MILPs—heavily depends on (P1) which cuts should be preferred, and (P2) how many cuts should be selected.
Revised: Cut selection—which aims to select a proper subset of the candidate cuts to improve the efficiency of solving MILPs—heavily depends on (P1) which cuts should be preferred, and (P2) how many cuts should be selected.

Original: Although many modern
Revised: Although many modern

Original: MILP solvers tackle (P1)-(P2) by manually designed heuristics, machine learning offers a promising approach to learn more effective heuristics from MILPs collected from specific applications.
Revised: MILP solvers tackle (P1)-(P2) by manually designed heuristics, machine learning offers a promising approach to learn more effective heuristics from MILPs collected from specific applications.

Original: However, many existing learning-based methods focus on learning which cuts should be preferred, neglecting the importance of learning the number of cuts that should be selected.
Revised: However, many existing learning-based methods focus on learning which cuts should be preferred, neglecting the importance of learning the number of cuts that should be selected.

Original: Moreover, we observe from extensive empirical results that (P3) what order of selected cuts should be preferred has a significant impact on the efficiency of solving MILPs as well.
Revised: Moreover, we observe from extensive empirical results that (P3) what order of selected cuts should be preferred has a significant impact on the efficiency of solving MILPs as well.

Original: To address this challenge, we propose a novel h ierarchical s e quence m odel (HEM) to learn cut selection policies via reinforcement learning.
Revised: To address this challenge, we propose a novel h ierarchical s e quence m odel (HEM) to learn cut selection policies via reinforcement learning.

Original: Specifically, HEM consists of a two-level model: (1) a higher-level model to learn the number of cuts that should be selected, (2) and a lower-level model—that formulates the cut selection task as a sequence to sequence learning problem—to learn policies selecting an ordered subset with the size determined by the higher-level model.
Revised: Specifically, HEM consists of a two-level model: (1) a higher-level model to learn the number of cuts that should be selected, (2) and a lower-level model—that formulates the cut selection task as a sequence to sequence learning problem—to learn policies selecting an ordered subset with the size determined by the higher-level model.

Original: To the best of our knowledge, HEM is the first method that can tackle (P1)-(P3) in cut selection simultaneously from a data-driven perspective.
Revised: To the best of our knowledge, HEM is the first method that can tackle (P1)-(P3) in cut selection simultaneously from a data-driven perspective.

Original: Experiments show that HEM significantly improves the efficiency of solving MILPs compared to human-designed and learning-based baselines on both synthetic and large-scale real-world MILPs, including MIPLIB 2017.
Revised: Experiments show that HEM significantly improves the efficiency of solving MILPs compared to human-designed and learning-based baselines on both synthetic and large-scale real-world MILPs, including MIPLIB 2017.

Original: Moreover, experiments demonstrate that HEM well generalizes to MILPs that are significantly larger than those seen during training.
Revised: Moreover, experiments demonstrate that HEM well generalizes to MILPs that are significantly larger than those seen during training.

Original:  I NTRODUCTION
Revised:  I NTRODUCTION

Original: Mixed-integer linear programming (MILP) is a general optimization formulation for a wide range of important real-world applications, such as supply chain management (Paschos, 2014), production planning (J¨unger et al., 2009), scheduling (Chen, 2010), facility location (Farahani & Hekmatfar, 2009), bin packing (Nair et al., 2020), etc.
Revised: Mixed-integer linear programming (MILP) is a general optimization formulation for a wide range of important real-world applications, such as supply chain management (Paschos, 2014), production planning (J¨unger et al., 2009), scheduling (Chen, 2010), facility location (Farahani & Hekmatfar, 2009), bin packing (Nair et al., 2020), etc.

Original: A standard MILP takes the form of z ∗ ≜
Revised: A standard MILP takes the form of z ∗ ≜

Original: min x { c ⊤ x | Ax ≤ b , x ∈ R n , x j ∈ Z for all j ∈ I} , (1) where c ∈ R n , A ∈ R m × n , b ∈ R m , x j denotes the j -th entry of vector x , I ⊆ { 1 , . . . , n } denotes the set of indices of integer variables, and z ∗ denotes the optimal objective value of the problem in (1).
Revised: min x { c ⊤ x | Ax ≤ b , x ∈ R n , x j ∈ Z for all j ∈ I} , (1) where c ∈ R n , A ∈ R m × n , b ∈ R m , x j denotes the j -th entry of vector x , I ⊆ { 1 , . . . , n } denotes the set of indices of integer variables, and z ∗ denotes the optimal objective value of the problem in (1).

Original: However, MILPs can be extremely hard to solve as they are NP -hard problems (Bengio et al., 2021).
Revised: However, MILPs can be extremely hard to solve as they are NP -hard problems (Bixby et al., 2004).

Original: To solve MILPs, many modern MILP solvers (Gurobi, 2021; Bestuzheva et al., 2021;
Revised: To solve MILPs, many modern MILP solvers (Gurobi, 2021; Bestuzheva et al., 2021;

Original: FICO Xpress, 2020) employ a branch-and-bound tree search algorithm (Land & Doig, 2010), in which a linear programming (LP) relaxation of a MILP (the problem in (1) or its subproblems) is solved at each node.
Revised: FICO Xpress, 2020) employ a branch-and-bound tree search algorithm (Land & Doig, 2010), in which a linear programming (LP) relaxation of a MILP (the problem in (1) or its subproblems) is solved at each node.

Original: To further enhance the performance of the tree search algorithm, cutting planes (cuts) (Gomory, 1960) are introduced to tighten the LP relaxations (Achterberg, 2007; Bengio et al., 2021).
Revised: To further enhance the performance of the tree search algorithm, cutting planes (cuts) (Gomory, 1960) are introduced to tighten the LP relaxations (Achterberg, 2007; Bengio et al., 2021).

Original: Existing work on cuts falls into two categories: cut generation and cut selection (Turner et al., 2022).
Revised: Existing work on cuts falls into two categories: cut generation and cut selection (Turner et al., 2022).

Original: Cut generation aims to generate cuts, i.e., valid linear inequalities that tighten the LP relaxations (Achterberg, 2007).
Revised: Cut generation aims to generate cuts, i.e., valid linear inequalities that tighten the LP relaxations (Achterberg, 2007).

Original: However, adding all the generated cuts to the LP relaxations can pose a computational problem (Wesselmann & Stuhl, 2012).
Revised: However, adding all the generated cuts to the LP relaxations can pose a computational problem (Wesselmann & Stuhl, 2012).

Original: To further improve the efficiency of solving MILPs, cut selection is proposed to select a proper subset of the generated cuts (Wesselmann & Stuhl, 2012).
Revised: To further improve the efficiency of solving MILPs, cut selection is proposed to select a proper subset of the generated cuts (Wesselmann & Stuhl, 2012).

Original: In this paper, we focus on the cut selection problem , which has a significant impact on the overall solver performance (Achterberg, 2007; Tang et al., 2020; Paulus et al., 2022).
Revised: In this paper, we focus on the cut selection problem , which has a significant impact on the overall solver performance (Achterberg, 2007; Tang et al., 2020; Paulus et al., 2022).

Original: Cut selection heavily depends on (P1) which cuts should be preferred, and (P2) how many cuts should be selected (Achterberg, 2007; Dey & Molinaro, 2018b).
Revised: Cut selection heavily depends on (P1) which cuts should be preferred, and (P2) how many cuts should be selected (Achterberg, 2007; Dey & Molinaro, 2018b).

Original: Many modern MILP solvers (Gurobi, 2021; Bestuzheva et al., 2021; FICO Xpress, 2020) tackle (P1)-(P2) by hard-coded heuristics designed by experts.
Revised: Many modern MILP solvers (Gurobi, 2021; Bestuzheva et al., 2021; FICO Xpress, 2020) tackle (P1)-(P2) by hard-coded heuristics designed by experts.

Original: However, hard-coded heuristics do not take into account underlying patterns among MILPs collected from certain types of real-world applications, e.g., day-to-day production planning, bin packing, and vehicle routing problems (Pochet & Wolsey, 2006; Laporte, 2009;
Revised: However, hard-coded heuristics do not take into account underlying patterns among MILPs collected from certain types of real-world applications, e.g., day-to-day production planning, bin packing, and vehicle routing problems (Pochet & Wolsey, 2006; Laporte, 2009;

Original: Nair et al., 2020).
Revised: Nair et al., 2020).

Original: To further improve the efficiency of MILP solvers, recent methods (Tang et al., 2020; Paulus et al., 2022; Huang et al., 2022) propose to learn cut selection policies via machine learning, especially reinforcement learning.
Revised: To further improve the efficiency of MILP solvers, recent methods (Tang et al., 2020; Paulus et al., 2022; Huang et al., 2022) propose to learn cut selection policies via machine learning, especially reinforcement learning.

Original: They offer promising approaches to learn more effective heuristics by capturing underlying patterns among MILPs from specific applications (Bengio et al., 2021).
Revised: They offer promising approaches to learn more effective heuristics by capturing underlying patterns among MILPs from specific applications (Bengio et al., 2021).

Original: However, many existing learning-based methods (Tang et al., 2020; Paulus et al., 2022; Huang et al., 2022)—which learn a scoring function to measure cut quality and select a fixed ratio/number of cuts with high scores—suffer from two limitations.
Revised: However, many existing learning-based methods (Tang et al., 2020; Paulus et al., 2022; Huang et al., 2022)—which learn a scoring function to measure cut quality and select a fixed ratio/number of cuts with high scores—suffer from two limitations.

Original: First, they learn which cuts should be preferred by learning a scoring function, neglecting the importance of learning the number of cuts that should be selected (Dey & Molinaro, 2018b).
Revised: First, they learn which cuts should be preferred by learning a scoring function, neglecting the importance of learning the number of cuts that should be selected (Dey & Molinaro, 2018b).

Original: Moreover, we observe from extensive empirical results that (P3) what order of selected cuts should be preferred significantly impacts the efficiency of solving MILPs as well (see Section 3).
Revised: Moreover, we observe from extensive empirical results that (P3) what order of selected cuts should be preferred significantly impacts the efficiency of solving MILPs as well (see Section 3).

Original: Second, they do not take into account the interaction among cuts when learning which cuts should be preferred, as they score each cut independently .
Revised: Second, they do not take into account the interaction among cuts when learning which cuts should be preferred, as they score each cut independently .

Original: As a result, they struggle to select cuts that complement each other nicely, which could severely hinder the efficiency of solving MILPs (Dey & Molinaro, 2018b).
Revised: As a result, they struggle to select cuts that complement each other nicely, which could severely hinder the efficiency of solving MILPs (Dey & Molinaro, 2018b).

Original: Indeed, we empirically show that they tend to select many similar cuts with high scores (see Experiment 4 in Section 5).
Revised: Indeed, we empirically show that they tend to select many similar cuts with high scores (see Experiment 4 in Section 5).

Original: To address the aforementioned challenges, we propose a novel h ierarchical s e quence m odel (HEM) to learn cut selection policies via reinforcement learning.
Revised: To address the aforementioned challenges, we propose a novel h ierarchical s e quence m odel (HEM) to learn cut selection policies via reinforcement learning.

Original: To the best of our knowledge, HEM is the first learning-based method that can tackle (P1)-(P3) simultaneously by proposing a two-level model.
Revised: To the best of our knowledge, HEM is the first learning-based method that can tackle (P1)-(P3) simultaneously by proposing a two-level model.

Original: Specifically, HEM is comprised of (1) a higher-level model to learn the number of cuts that should be selected, (2) and a lower-level model to learn policies selecting an ordered subset with the size determined by the higher-level model.
Revised: Specifically, HEM is comprised of (1) a higher-level model to learn the number of cuts that should be selected, (2) and a lower-level model to learn policies selecting an ordered subset with the size determined by the higher-level model.

Original: The lower-level model formulates the cut selection task as a sequence to sequence learning problem, leading to two major advantages.
Revised: The lower-level model formulates the cut selection task as a sequence to sequence learning problem, leading to two major advantages.

Original: First, the sequence model is popular in capturing the underlying order information (Vinyals et al., 2016), which is critical for tackling (P3) .
Revised: First, the sequence model is popular in capturing the underlying order information (Vinyals et al., 2016), which is critical for tackling (P3) .

Original: Second, the sequence model can well capture the interaction among cuts, as it models the joint conditional probability of the selected cuts given an input sequence of the candidate cuts.
Revised: Second, the sequence model can well capture the interaction among cuts, as it models the joint conditional probability of the selected cuts given an input sequence of the candidate cuts.

Original: As a result, experiments show that HEM significantly outperforms human-designed and learning-based baselines in terms of solving efficiency on three synthetic MILP problems and seven challenging MILP problems.
Revised: As a result, experiments show that HEM significantly outperforms human-designed and learning-based baselines in terms of solving efficiency on three synthetic MILP problems and seven challenging MILP problems.

Original: The challenging MILP problems include some benchmarks from
Revised: The challenging MILP problems include some benchmarks from

Original: MIPLIB 2017 (Gleixner et al., 2021) and large-scale real-world production planning problems.
Revised: MIPLIB 2017 (Gleixner et al., 2021) and large-scale real-world production planning problems.

Original: Our results demonstrate the strong ability to enhance modern MILP solvers with our proposed HEM in real-world applications.
Revised: Our results demonstrate the strong ability to enhance modern MILP solvers with our proposed HEM in real-world applications.

Original: Moreover, experiments demonstrate that HEM can well generalize to
Revised: Moreover, experiments demonstrate that HEM can well generalize to

Original: MILPs that are significantly larger than those seen during training.
Revised: MILPs that are significantly larger than those seen during training.

Original: We summarize our major contributions as follows.
Revised: We summarize our major contributions as follows.

Original: (1) We observe from extensive empirical results that the order of selected cuts has a significant impact on the efficiency of solving MILPs (see Section 3).
Revised: (1) We observe from extensive empirical results that the order of selected cuts has a significant impact on the efficiency of solving MILPs (see Section 3).

Original: (2) To the best of our knowledge, our proposed HEM is the first method that is able to tackle (P1)-(P3) in cut selection simultaneously from a data-driven perspective.
Revised: (2) To the best of our knowledge, our proposed HEM is the first method that is able to tackle (P1)-(P3) in cut selection simultaneously from a data-driven perspective.

Original: (3) We propose to formulate the cut selection task as a sequence to sequence learning problem, which not only can capture the underlying order information, but also well captures the interaction among cuts to select cuts that complement each other nicely.
Revised: (3) We propose to formulate the cut selection task as a sequence to sequence learning problem, which not only can capture the underlying order information, but also well captures the interaction among cuts to select cuts that complement each other nicely.

Original: (4) Experiments demonstrate that HEM achieves significant improvements over competitive baselines on challenging MILP problems, including some benchmarks from MIPLIB 2017 and large-scale real-world production planning problems.
Revised: (4) Experiments demonstrate that HEM achieves significant improvements over competitive baselines on challenging MILP problems, including some benchmarks from MIPLIB 2017 and large-scale real-world production planning problems.

Original:  B ACKGROUND
Revised:  B ACKGROUND

Original: Cutting planes.
Revised: Cutting planes.

Original: Given the MILP problem in (1), we drop all its integer constraints to obtain its linear programming (LP) relaxation , which takes the form of
Revised: Given the MILP problem in (1), we drop all its integer constraints to obtain its linear programming (LP) relaxation , which takes the form of

Original: [Equation]
Revised: [Equation]

Original: Since the problem in (2) expands the feasible set of the problem in (1), we have z ∗ LP ≤ z ∗ .
Revised: Since the problem in (2) expands the feasible set of the problem in (1), we have z ∗ LP ≤ z ∗ .

Original: We denote any lower bound found via an LP relaxation by a dual bound .
Revised: We denote any lower bound found via an LP relaxation by a dual bound .

Original: Given the LP relaxation in (2), cutting planes (cuts) are linear inequalities that are added to the LP relaxation in the attempt to tighten it without removing any integer feasible solutions of the problem in (1).
Revised: Given the LP relaxation in (2), cutting planes (cuts) are linear inequalities that are added to the LP relaxation in the attempt to tighten it without removing any integer feasible solutions of the problem in (1).

Original: Cuts generated by MILP solvers are added in successive rounds.
Revised: Cuts generated by MILP solvers are added in successive rounds.

Original: Specifically, each round k involves (i) solving the current LP relaxation, (ii) generating a pool of candidate cuts C k , (iii) selecting a subset S k ⊆ C k , (iv) adding S k to the current LP relaxation to obtain the next LP relaxation, (v) and proceeding to the next round.
Revised: Specifically, each round k involves (i) solving the current LP relaxation, (ii) generating a pool of candidate cuts C k , (iii) selecting a subset S k ⊆ C k , (iv) adding S k to the current LP relaxation to obtain the next LP relaxation, (v) and proceeding to the next round.

Original: Adding all the generated cuts to the LP relaxation would maximally strengthen the LP relaxation and improve the lower bound at each round.
Revised: Adding all the generated cuts to the LP relaxation would maximally strengthen the LP relaxation and improve the lower bound at each round.

Original: However, adding too many cuts could lead to large models, which can increase the computational burden and present numerical instabilities (Wesselmann & Stuhl, 2012; Paulus et al., 2022).
Revised: However, adding too many cuts could lead to large models, which can increase the computational burden and present numerical instabilities (Wesselmann & Stuhl, 2012; Paulus et al., 2022).

Original: Therefore, cut selection is proposed to select a proper subset of the candidate cuts, which is significant for improving the efficiency of solving
Revised: Therefore, cut selection is proposed to select a proper subset of the candidate cuts, which is significant for improving the efficiency of solving

Original: MILPs (Dey & Molinaro, 2018b; Tang et al., 2020).
Revised: MILPs (Dey & Molinaro, 2018b; Tang et al., 2020).

Original: Branch-and-cut.
Revised: Branch-and-cut.

Original: In modern MILP solvers, cutting planes are often combined with the branch-andbound algorithm (Land & Doig, 2010), which is known as the branch-and-cut algorithm (Bengio et al., 2021).
Revised: In modern MILP solvers, cutting planes are often combined with the branch-andbound algorithm (Land & Doig, 2010), which is known as the branch-and-cut algorithm (Mitchell, 2002).

Original: Branch-and-bound techniques perform implicit enumeration by building a search tree, in which every node represents a subproblem of the original problem in (1).
Revised: Branch-and-bound techniques perform implicit enumeration by building a search tree, in which every node represents a subproblem of the original problem in (1).

Original: The solving process begins by selecting a leaf node of the tree and solving its LP relaxation.
Revised: The solving process begins by selecting a leaf node of the tree and solving its LP relaxation.

Original: Let x ∗ be the optimal solution of the LP relaxation.
Revised: Let x ∗ be the optimal solution of the LP relaxation.

Original: If x ∗ violates the original integrality constraints, two subproblems (child nodes) of the leaf node are created by branching .
Revised: If x ∗ violates the original integrality constraints, two subproblems (child nodes) of the leaf node are created by branching .

Original: Specifically, the leaf node is added with constraints x i ≤ ⌊ x ∗ i ⌋ and x i ≥ ⌈ x ∗ i ⌉ , respectively, where x i denotes the i -th variable, x ∗ i denotes the i -th entry of vector x ∗ , and ⌊⌋ and ⌈⌉ denote the floor and ceil functions.
Revised: Specifically, the leaf node is added with constraints x i ≤ ⌊ x ∗ i ⌋ and x i ≥ ⌈ x ∗ i ⌉ , respectively, where x i denotes the i -th variable, x ∗ i denotes the i -th entry of vector x ∗ , and ⌊⌋ and ⌈⌉ denote the floor and ceil functions.

Original: In contrast, if x ∗ is a (mixed-)integer solution of (1), then we obtain an upper bound on the optimal objective value of (1), which we denote by primal bound .
Revised: In contrast, if x ∗ is a (mixed-)integer solution of (1), then we obtain an upper bound on the optimal objective value of (1), which we denote by primal bound .

Original: In modern MILP solvers, the addition of cutting planes is alternated with the branching phase.
Revised: In modern MILP solvers, the addition of cutting planes is alternated with the branching phase.

Original: That is, cuts are added at search tree nodes before branching to tighten their LP relaxations.
Revised: That is, cuts are added at search tree nodes before branching to tighten their LP relaxations.

Original: Since strengthening the relaxation before starting to branch is decisive to ensure an efficient tree search (Wesselmann & Stuhl, 2012; Bengio et al., 2021), we focus on only adding cuts at the root node, which follows Gasse et al.
Revised: Since strengthening the relaxation before starting to branch is decisive to ensure an efficient tree search (Wesselmann & Stuhl, 2012; Bengio et al., 2021), we focus on only adding cuts at the root node, which follows Gasse et al.

Original: ; Paulus et al. (2022).
Revised: ; Paulus et al. (2022).

Original: Primal-dual gap integral.
Revised: Primal-dual gap integral.

Original: We keep track of two important bounds when running branch-and-cut, i.e., the global primal and dual bounds, which are the best upper and lower bounds on the optimal objective value of (1), respectively.
Revised: We keep track of two important bounds when running branch-and-cut, i.e., the global primal and dual bounds, which are the best upper and lower bounds on the optimal objective value of (1), respectively.

Original: We define the primal-dual gap integral (PD integral) by the area between the curve of the solver’s global primal bound and the curve of the solver’s global dual bound.
Revised: We define the primal-dual gap integral (PD integral) by the area between the curve of the solver’s global primal bound and the curve of the solver’s global dual bound.

Original: The PD integral is a widely used metric for evaluating solver performance (Bowly et al., 2021; Cao et al., 2022).
Revised: The PD integral is a widely used metric for evaluating solver performance (Bowly et al., 2021; Cao et al., 2022).

Original: We provide more details in Appendix C.1.
Revised: We provide more details in Appendix C.1.

Original:  M OTIVATING R ESULTS
Revised:  M OTIVATING R ESULTS

Original: [Figure]
Revised: [Figure]

Revised: [Figure]

Original: [Figure]
Revised: [Figure]

Original:  cuts  cuts  to  LP  in order
Revised: We empirically show that the order of selected cuts , i.e., the selected cuts are added to the LP relaxations in this order, significantly impacts the efficiency of solving MILPs.

Original: MILPs ratio of selected cuts matters significantly  MILPs (see Appendix G.1)
Revised: Moreover, we empirically show that the ratio of selected cuts matters significantly when solving MILPs (see Appendix G.1).

Original:  Appendix D
Revised: Please see Appendix D. details of the datasets used in this section.

Original: Order matters.
Revised: Order matters.

Original: Previous work (Bixby, 1992; Maros, 2002; Li et al., 2022) has shown that the order of constraints for a given linear program (LP) significantly impacts its constructed initial basis, which is important for solving the LP.
Revised: Previous work (Bixby, 1992; Maros, 2002; Li et al., 2022) has shown that the order of constraints for a given linear program (LP) significantly impacts its constructed initial basis, which is important for solving the LP.

Original: As a cut is a linear constraint, adding cuts to the LP relaxations is equivalent to adding constraints to the LP relaxations.
Revised: As a cut is a linear constraint, adding cuts to the LP relaxations is equivalent to adding constraints to the LP relaxations.

Original: Therefore, the order of added cuts could have a significant impact on solving the LP relaxations as well, thus being important for solving MILPs.
Revised: Therefore, the order of added cuts could have a significant impact on solving the LP relaxations as well, thus being important for solving MILPs.

Original: Indeed, our empirical results show that this is the case.
Revised: Indeed, our empirical results show that this is the case.

Original: (1) We design a RandomAll cut selection rule, which randomly permutes all the candidate cuts, and adds all the cuts to the LP relaxations in the random order.
Revised: (1) We design a RandomAll cut selection rule, which randomly permutes all the candidate cuts, and adds all the cuts to the LP relaxations in the random order.

Original: We evaluate RandomAll on five challenging datasets, namely D1, D2, D3, D4, and D5.
Revised: We evaluate RandomAll on five challenging datasets, namely D1, D2, D3, D4, and D5.

Original: We use the SCIP 8.0.0 (Bestuzheva et al.,
Revised: We use the SCIP 8.0.0 (Bestuzheva et al.,

Original: 2021) as the backend solver, and evaluate the solver performance by the average PD integral within a time limit.
Revised: 2021) as the backend solver, and evaluate the solver performance by the average PD integral within a time limit.

Original: We evaluate RandomAll on each dataset over ten random seeds, and each bar in Figure 1a shows the mean and standard deviation (stdev) of its performance on each dataset.
Revised: We evaluate RandomAll on each dataset over ten random seeds, and each bar in Figure 1a shows the mean and standard deviation (stdev) of its performance on each dataset.

Original: As shown in
Revised: As shown in

Original: Figure 1a, the performance of RandomAll on each dataset varies widely with the order of selected cuts.
Revised: Figure 1a, the performance of RandomAll on each dataset varies widely with the order of selected cuts.

Original: (2) We further design a RandomNV cut selection rule.
Revised: (2) We further design a RandomNV cut selection rule.

Original: RandomNV is different from RandomAll in that it selects a given ratio of the candidate cuts rather than all the cuts.
Revised: RandomNV is different from RandomAll in that it selects a given ratio of the candidate cuts rather than all the cuts.

Original: RandomNV first scores each cut using the Normalized Violation (Huang et al., 2022) and selects a given ratio of cuts with high scores.
Revised: RandomNV first scores each cut using the Normalized Violation (Huang et al., 2022) and selects a given ratio of cuts with high scores.

Original: It then randomly permutes the selected cuts.
Revised: It then randomly permutes the selected cuts.

Original: Each bar in Figure 1b shows the mean and stdev of the performance of RandomNV with a given ratio on the same dataset.
Revised: Each bar in Figure 1b shows the mean and stdev of the performance of RandomNV with a given ratio on the same dataset.

Original: Figures 1a and 1b show that adding the same selected cuts in different order leads to variable solver performance, which demonstrates that the order of selected cuts is important for solving MILPs.
Revised: Figures 1a and 1b show that adding the same selected cuts in different order leads to variable solver performance, which demonstrates that the order of selected cuts is important for solving MILPs.

Original:  L EARNING C UT S ELECTION VIA H IERARCHICAL S EQUENCE M ODEL
Revised:  L EARNING C UT S ELECTION VIA H IERARCHICAL S EQUENCE M ODEL

Original: In the cut selection task, the optimal subsets that should be selected are inaccessible, but one can assess the quality of selected subsets using a solver and provide the feedbacks to learning algorithms.
Revised: In the cut selection task, the optimal subsets that should be selected are inaccessible, but one can assess the quality of selected subsets using a solver and provide the feedbacks to learning algorithms.

Original: Therefore, we leverage reinforcement learning (RL) to learn cut selection policies.
Revised: Therefore, we leverage reinforcement learning (RL) to learn cut selection policies.

Original: In this section, we provide a detailed description of our proposed RL framework for learning cut selection.
Revised: In this section, we provide a detailed description of our proposed RL framework for learning cut selection.

Original: First, we present our formulation of the cut selection as a Markov decision process (MDP) (Sutton & Barto, 2018).
Revised: First, we present our formulation of the cut selection as a Markov decision process (MDP) (Sutton & Barto, 2018).

Original: Then, we present a detailed description of our proposed h ierarchical s e quence m odel (HEM).
Revised: Then, we present a detailed description of our proposed h ierarchical s e quence m odel (HEM).

Original: Finally, we derive a hierarchical policy gradient for training HEM efficiently.
Revised: Finally, we derive a hierarchical policy gradient for training HEM efficiently.

Original: [Equation]
Revised: [Equation]

Original: [Figure]
Revised: [Figure]

Original: 𝑠𝑡𝑎𝑡𝑒 𝑠
Revised: 𝑠𝑡𝑎𝑡𝑒 𝑠

Original: Reinforcement Learning Formulation
Revised: Reinforcement Learning Formulation

Original: As shown in Figure 2, we formulate a MILP solver as the environment and our proposed
Revised: As shown in Figure 2, we formulate a MILP solver as the environment and our proposed

Original: HEM as the agent.
Revised: HEM as the agent.

Original: We consider an MDP defined by the tuple ( S , A , r, f ) .
Revised: We consider an MDP defined by the tuple ( S , A , r, f ) .

Original: Specifically, we specify the state space S , the action space A , the reward function r : S ×
Revised: Specifically, we specify the state space S , the action space A , the reward function r : S ×

Original: A → R , the transition function f , and the terminal state in the following.
Revised: A → R , the transition function f , and the terminal state in the following.

Original: (1) The state space S .
Revised: (1) The state space S .

Original: Since the current LP relaxation and the generated cuts contain the core information for cut selection, we define a state s by ( M LP , C , x ∗ LP ) .
Revised: Since the current LP relaxation and the generated cuts contain the core information for cut selection, we define a state s by ( M LP , C , x ∗ LP ) .

Original: Here M LP denotes the mathematical model of the current LP relaxation, C denotes the set of the candidate cuts, and x ∗ LP denotes the optimal solution of the LP relaxation.
Revised: Here M LP denotes the mathematical model of the current LP relaxation, C denotes the set of the candidate cuts, and x ∗ LP denotes the optimal solution of the LP relaxation.

Original: To encode the state information, we follow Achterberg (2007); Huang et al.
Revised: To encode the state information, we follow Achterberg (2007); Huang et al.

Original: (2022) to design thirteen features for each candidate cut based on the information of ( M LP , C , x ∗ LP ) .
Revised: (2022) to design thirteen features for each candidate cut based on the information of ( M LP , C , x ∗ LP ) .

Original: That is, we actually represent a state s by a sequence of thirteen-dimensional feature vectors .
Revised: That is, we actually represent a state s by a sequence of thirteen-dimensional feature vectors .

Original: We present details of the designed features in Appendix
Revised: We present details of the designed features in Appendix

Original: F.1.
Revised: F.1.

Original: The action space A .
Revised: The action space A .

Original: To take into account the ratio and order of selected cuts, we define the action space by all the ordered subsets of the candidate cuts C .
Revised: To take into account the ratio and order of selected cuts, we define the action space by all the ordered subsets of the candidate cuts C .

Original: It can be challenging to explore the action space efficiently, as the cardinality of the action space can be extremely large due to its combinatorial structure.
Revised: It can be challenging to explore the action space efficiently, as the cardinality of the action space can be extremely large due to its combinatorial structure.

Original: The reward function r .
Revised: The reward function r .

Original: To evaluate the impact of the added cuts on solving MILPs, we design the reward function by (i) measures collected at the end of solving LP relaxations such as the dual bound improvement, (ii) or end-of-run statistics, such as the solving time and the primal-dual gap integral.
Revised: To evaluate the impact of the added cuts on solving MILPs, we design the reward function by (i) measures collected at the end of solving LP relaxations such as the dual bound improvement, (ii) or end-of-run statistics, such as the solving time and the primal-dual gap integral.

Original: For the first, the reward r ( s, a ) can be defined as the negative dual bound improvement at each step.
Revised: For the first, the reward r ( s, a ) can be defined as the negative dual bound improvement at each step.

Original: For the second, the reward r ( s, a ) can be defined as zero except for the last step ( s T , a T ) in a trajectory, i.e., r ( s T , a T ) is defined by the negative solving time or the negative primal-dual gap integral.
Revised: For the second, the reward r ( s, a ) can be defined as zero except for the last step ( s T , a T ) in a trajectory, i.e., r ( s T , a T ) is defined by the negative solving time or the negative primal-dual gap integral.

Original: The transition function f .
Revised: The transition function f .

Original: The transition function maps the current state s and the action a to the next state s ′ , where s ′ represents the next LP relaxation generated by adding the selected cuts at the current LP relaxation.
Revised: The transition function maps the current state s and the action a to the next state s ′ , where s ′ represents the next LP relaxation generated by adding the selected cuts at the current LP relaxation.

Original: The terminal state.
Revised: The terminal state.

Original: There is no standard and unified criterion to determine when to terminate the cut separation procedure (Paulus et al., 2022).
Revised: There is no standard and unified criterion to determine when to terminate the cut separation procedure (Paulus et al., 2022).

Original: In this paper, we set the cut separation roundsT as one following Huang et al. That is, we terminate the cut separation after one round.
Revised: Suppose we set the cut separation rounds as T , then the solver environment terminates the cut separation after T rounds.

Original: Nevertheless, our proposed RL framework can be easily applied to the multiple rounds setting, i.e., T > 1 . If T = 1 , then the decision problem is a one-step decision problem. More precisely, it is a contextual bandit (Sutton & Barto, 2018).
Revised: Under the multiple rounds setting (i.e., T > 1 ), we formulate the cut selection as a Markov decision process. Under the one round setting (i.e., T = 1 ), the formulation can be simplified as a contextual bandit.

Original: Hierarchical Sequence Model
Revised: Hierarchical Sequence Model

Original: Motivation.
Revised: Motivation.

Original: Let π denote the cut selection policy π : S → P ( A ) , where P ( A ) denotes the probability distribution over the action space, and π ( ·| s ) denotes the probability distribution over the action space given the state s .
Revised: Let π denote the cut selection policy π : S → P ( A ) , where P ( A ) denotes the probability distribution over the action space, and π ( ·| s ) denotes the probability distribution over the action space given the state s .

Original: We emphasize that learning such policies can tackle (P1)-(P3) in cut selection simultaneously.
Revised: We emphasize that learning such policies can tackle (P1)-(P3) in cut selection simultaneously.

Original: However, directly learning such policies is challenging for the following reasons.
Revised: However, directly learning such policies is challenging for the following reasons.

Original: First, it is challenging to explore the action space efficiently, as the cardinality of the action space can be extremely large due to its combinatorial structure.
Revised: First, it is challenging to explore the action space efficiently, as the cardinality of the action space can be extremely large due to its combinatorial structure.

Original: Second, the length and max length of actions (i.e., ordered subsets) are variable across different MILPs.
Revised: Second, the length and max length of actions (i.e., ordered subsets) are variable across different MILPs.

Original: However, traditional RL usually deals with problems whose actions have a fixed length.
Revised: However, traditional RL usually deals with problems whose actions have a fixed length.

Original: Instead of directly learning the aforementioned policy, many existing learning-based methods (Tang et al., 2020; Huang et al., 2022; Paulus et al., 2022) learn a scoring function that outputs a score given a cut, and select a fixed ratio/number of cuts with high scores.
Revised: Instead of directly learning the aforementioned policy, many existing learning-based methods (Tang et al., 2020; Huang et al., 2022; Paulus et al., 2022) learn a scoring function that outputs a score given a cut, and select a fixed ratio/number of cuts with high scores.

Original: However, they suffer from two limitations as mentioned in Section 1.
Revised: However, they suffer from two limitations as mentioned in Section 1.

Original: Policy network architecture.
Revised: Policy network architecture.

Original: To tackle the aforementioned problems, we propose a novel hierarchical sequence model (HEM) to learn cut selection policies.
Revised: To tackle the aforementioned problems, we propose a novel hierarchical sequence model (HEM) to learn cut selection policies.

Original: To promote efficient exploration,
Revised: To promote efficient exploration,

Original: HEM leverages the hierarchical structure of the cut selection task to decompose the policy into two sub-policies, i.e., a higher-level policy π h and a lower-level policy π l .
Revised: HEM leverages the hierarchical structure of the cut selection task to decompose the policy into two sub-policies, i.e., a higher-level policy π h and a lower-level policy π l .

Original: The policy network architecture of HEM is also illustrated in Figure 2.
Revised: The policy network architecture of HEM is also illustrated in Figure 2.

Original: First , the higher-level policy learns the number of cuts that should be selected by predicting a proper ratio.
Revised: First , the higher-level policy learns the number of cuts that should be selected by predicting a proper ratio.

Original: Suppose the length of the state is N and the predicted ratio is k , then the predicted number of cuts that should be selected is ⌊ N ∗ k ⌋ , where ⌊·⌋ denotes the floor function.
Revised: Suppose the length of the state is N and the predicted ratio is k , then the predicted number of cuts that should be selected is ⌊ N ∗ k ⌋ , where ⌊·⌋ denotes the floor function.

Original: We define the higher-level policy by π h : S → P ([0 , 1]) , where π h ( ·| s ) denotes the probability distribution over [0 , 1] given the state s .
Revised: We define the higher-level policy by π h : S → P ([0 , 1]) , where π h ( ·| s ) denotes the probability distribution over [0 , 1] given the state s .

Original: Second , the lower-level policy learns to select an ordered subset with the size 1 determined by the higher-level policy.
Revised: Second , the lower-level policy learns to select an ordered subset with the size 1 determined by the higher-level policy.

Original: We define the lower-level policy by π l : S × [0 , 1] → P ( A ) , where π l ( ·| s, k ) denotes the probability distribution over the action space given the state s and the ratio k .
Revised: We define the lower-level policy by π l : S × [0 , 1] → P ( A ) , where π l ( ·| s, k ) denotes the probability distribution over the action space given the state s and the ratio k .

Original: Specifically, we formulate the lower-level policy as a sequence model, which can capture the interaction among cuts.
Revised: Specifically, we formulate the lower-level policy as a sequence model, which can capture the interaction among cuts.

Original: Finally , we derive the cut selection policy via the law of total probability, i.e., π ( a k | s ) =
Revised: Finally , we derive the cut selection policy via the law of total probability, i.e., π ( a k | s ) =

Original: E k ∼ π h ( ·| s )
Revised: E k ∼ π h ( ·| s )

Original: [ π l ( a k | s, k )] , where k denotes the given ratio and a k denotes the action.
Revised: [ π l ( a k | s, k )] , where k denotes the given ratio and a k denotes the action.

Original: The policy is computed by an expectation, as a k cannot determine the ratio k .
Revised: The policy is computed by an expectation, as a k cannot determine the ratio k .

Original: For example, suppose that N = 100 and the length of a k is 10 , then the ratio k can be any number in the interval [0 . 1 , 0 . 11) .
Revised: For example, suppose that N = 100 and the length of a k is 10 , then the ratio k can be any number in the interval [0 . 1 , 0 . 11) .

Original: Actually, we sample an action from the policy π by first sampling a ratio k from π h and then sampling an action from π l given the ratio.
Revised: Actually, we sample an action from the policy π by first sampling a ratio k from π h and then sampling an action from π l given the ratio.

Original: For the higher-level policy, we first model the higher-level policy as a tanh-Gaussian, i.e., a Gaussian distribution with an invertible squashing function ( tanh ), which is commonly used in deep reinforcement learning (Schulman et al., 2017; Haarnoja et al., 2018).
Revised: For the higher-level policy, we first model the higher-level policy as a tanh-Gaussian, i.e., a Gaussian distribution with an invertible squashing function ( tanh ), which is commonly used in deep reinforcement learning (Schulman et al., 2017; Haarnoja et al., 2018).

Original: The mean and variance of the Gaussian are given by neural networks.
Revised: The mean and variance of the Gaussian are given by neural networks.

Original: The support of the tanh-Gaussian is [ − 1 , 1] , but a ratio of selected cuts should belong to [0 , 1] .
Revised: The support of the tanh-Gaussian is [ − 1 , 1] , but a ratio of selected cuts should belong to [0 , 1] .

Original: Thus, we further perform a linear transformation on the tanh-Gaussian.
Revised: Thus, we further perform a linear transformation on the tanh-Gaussian.

Original: Specifically, we define the parameterized higher-level policy by π hθ 1 ( ·| s )
Revised: Specifically, we define the parameterized higher-level policy by π hθ 1 ( ·| s )

Original: = 0 .
Revised: = 0 .

Original: 5 ∗ tanh ( K )
Revised: 5 ∗ tanh ( K )

Original: + 0 .
Revised: + 0 .

Original: 5 , where K ∼ N ( µ θ 1 ( s ) , σ θ 1 ( s )) .
Revised: 5 , where K ∼ N ( µ θ 1 ( s ) , σ θ 1 ( s )) .

Original: Since the sequence lengths of states are variable across different instances (MILPs), we use a long-short term memory (LSTM) (Hochreiter & Schmidhuber, 1997) network to embed the sequence of candidate cuts.
Revised: Since the sequence lengths of states are variable across different instances (MILPs), we use a long-short term memory (LSTM) (Hochreiter & Schmidhuber, 1997) network to embed the sequence of candidate cuts.

Original: We then use a multi-layer perceptron (MLP)
Revised: We then use a multi-layer perceptron (MLP)

Original: (Goodfellow et al., 2016) to predict the mean and variance from the last hidden state of the LSTM.
Revised: (Goodfellow et al., 2016) to predict the mean and variance from the last hidden state of the LSTM.

Original: For the lower-level policy, we formulate it as a sequence model.
Revised: For the lower-level policy, we formulate it as a sequence model.

Original: That is, its input is a sequence of candidate cuts, and its output is the probability distribution over ordered subsets of candidate cuts with the size determined by the higher-level policy.
Revised: That is, its input is a sequence of candidate cuts, and its output is the probability distribution over ordered subsets of candidate cuts with the size determined by the higher-level policy.

Original: Specifically, given a state action pair ( s, k, a k ) , the sequence model computes the conditional probability π lθ 2 ( a k | s, k ) using a parametric model to estimate the terms of the probability chain rule, i.e., π lθ 2 ( a k | s, k ) =
Revised: Specifically, given a state action pair ( s, k, a k ) , the sequence model computes the conditional probability π lθ 2 ( a k | s, k ) using a parametric model to estimate the terms of the probability chain rule, i.e., π lθ 2 ( a k | s, k ) =

Original: (cid:81) mi
Revised: (cid:81) mi

Original: =1 p θ 2 ( a ik | a 1 k , . .
Revised: =1 p θ 2 ( a ik | a 1 k , . .

Original: , a i − 1 k , s, k ) .
Revised: , a i − 1 k , s, k ) .

Original: Here s = { s 1 , . . .
Revised: Here s = { s 1 , . . .

Original: , s N } is the input sequence, m =
Revised: , s N } is the input sequence, m =

Original: ⌊ N ∗ k ⌋ is the length of the output sequence, and a k = { a 1 k , . . .
Revised: ⌊ N ∗ k ⌋ is the length of the output sequence, and a k = { a 1 k , . . .

Original: , a mk } is a sequence of m indices, each corresponding a position in the input sequence s .
Revised: , a mk } is a sequence of m indices, each corresponding a position in the input sequence s .

Original: Such policy can be parametrized by the vanilla sequence model commonly used in machine translation (Sutskever et al., 2014; Vaswani et al., 2017).
Revised: Such policy can be parametrized by the vanilla sequence model commonly used in machine translation (Sutskever et al., 2014; Vaswani et al., 2017).

Original: However, the vanilla sequence model can only be applied to learning on a single instance, as the number of candidate cuts varies on different instances.
Revised: However, the vanilla sequence model can only be applied to learning on a single instance, as the number of candidate cuts varies on different instances.

Original: To generalize across different instances, we use a pointer network (Vinyals et al., 2015; Bello* et al., 2017)—which uses attention as a pointer to select a member of the input sequence as the output at each decoder step—to parametrize π lθ(see Appendix F.4.1 for details).
Revised: To generalize across different instances, we use a pointer network (Vinyals et al., 2015; Bello* et al., 2017)—which uses attention as a pointer to select a member of the input sequence as the output at each decoder step—to parametrize π lθ(see Appendix F.4.1 for details).

Original: To the best of our knowledge, we are the first to formulate the cut selection task as a sequence to sequence learning problem and apply the pointer network to cut selection.
Revised: To the best of our knowledge, we are the first to formulate the cut selection task as a sequence to sequence learning problem and apply the pointer network to cut selection.

Original: This leads to two major advantages: (1) capturing the underlying order information, (2) and the interaction among cuts.
Revised: This leads to two major advantages: (1) capturing the underlying order information, (2) and the interaction among cuts.

Original: This is also illustrated through an example in Appendix E.
Revised: This is also illustrated through an example in Appendix E.

Original: Training: hierarchical policy gradient
Revised: Training: hierarchical policy gradient

Original: For the cut selection task, we aim to find θ that maximizes the expected reward over all trajectories
Revised: For the cut selection task, we aim to find θ that maximizes the expected reward over all trajectories

Original: [Equation]
Revised: [Equation]

Original: where θ =
Revised: where θ =

Original: [ θ 1 , θ 2 ] with [ θ 1 , θ 2 ] denoting the concatenation of the two vectors, π θ ( a k | s )
Revised: [ θ 1 , θ 2 ] with [ θ 1 , θ 2 ] denoting the concatenation of the two vectors, π θ ( a k | s )

Original: =
Revised: =

Original: E k ∼ π hθ 1 ( ·| s )
Revised: E k ∼ π hθ 1 ( ·| s )

Original: [ π lθ 2 ( a k | s, k )] , and µ denotes the initial state distribution.
Revised: [ π lθ 2 ( a k | s, k )] , and µ denotes the initial state distribution.

Original: To train the policy with a hierarchical structure, we derive a hierarchical policy gradient following the well-known policy gradient theorem (Sutton et al., 1999a; Sutton & Barto, 2018).
Revised: To train the policy with a hierarchical structure, we derive a hierarchical policy gradient following the well-known policy gradient theorem (Sutton et al., 1999a; Sutton & Barto, 2018).

Original: Proposition 1.
Revised: Proposition 1.

Original: Given the cut selection policy π θ ( a k | s ) =
Revised: Given the cut selection policy π θ ( a k | s ) =

Original: E k ∼ π hθ 1 ( ·| s )
Revised: E k ∼ π hθ 1 ( ·| s )

Original: [ π lθ 2 ( a k | s, k )] and the training objective (3), the hierarchical policy gradient takes the form of
Revised: [ π lθ 2 ( a k | s, k )] and the training objective (3), the hierarchical policy gradient takes the form of

Original: [Equation]
Revised: [Equation]

Original: We provide detailed proof in Appendix A.
Revised: We provide detailed proof in Appendix A.

Original: We use the derived hierarchical policy gradient to update the parameters of the higher-level and lower-level policies.
Revised: We use the derived hierarchical policy gradient to update the parameters of the higher-level and lower-level policies.

Original: We implement the training algorithm in a parallel manner that is closely related to the asynchronous advantage actor-critic (A3C) (Mnih et al., 2016).
Revised: We implement the training algorithm in a parallel manner that is closely related to the asynchronous advantage actor-critic (A3C) (Mnih et al., 2016).

Original: Due to limited space, we summarize the procedure of the training algorithm in Appendix
Revised: Due to limited space, we summarize the procedure of the training algorithm in Appendix

Original: F.3.6.
Revised: F.3.6.

Original: Moreover, we discuss some more advantages of HEM (see Appendix F.4.3 for details).
Revised: Moreover, we discuss some more advantages of HEM (see Appendix F.4.3 for details).

Original: HEM leverages the hierarchical structure of the cut selection task, which is important for efficient exploration in complex decision-making tasks (Sutton et al., 1999b).
Revised: HEM leverages the hierarchical structure of the cut selection task, which is important for efficient exploration in complex decision-making tasks (Sutton et al., 1999b).

Original: (2) We train HEM via gradientbased algorithms, which is sample efficient (Sutton & Barto, 2018).
Revised: (2) We train HEM via gradientbased algorithms, which is sample efficient (Sutton & Barto, 2018).

Original:  E XPERIMENTS
Revised:  E XPERIMENTS

Original: Our experiments have five main parts: Experiment 1.
Revised: Our experiments have five main parts: Experiment 1.

Original: Evaluate our approach on three classical
Revised: Evaluate our approach on three classical

Original: MILP problems and six challenging MILP problem benchmarks from diverse application areas.
Revised: MILP problems and six challenging MILP problem benchmarks from diverse application areas.

Original: Experiment 2.
Revised: Experiment 2.

Original: Perform carefully designed ablation studies to provide further insight into HEM.
Revised: Perform carefully designed ablation studies to provide further insight into HEM.

Original: Experiment 3.
Revised: Experiment 3.

Original: Test whether HEM can generalize to instances significantly larger than those seen during training.
Revised: Test whether HEM can generalize to instances significantly larger than those seen during training.

Original: Experiment 4.
Revised: Experiment 4.

Original: Visualize the cuts selected by our method compared to the baselines.
Revised: Visualize the cuts selected by our method compared to the baselines.

Original: Experiment 5.
Revised: Experiment 5.

Original: Deploy our approach to real-world production planning problems.
Revised: Deploy our approach to real-world production planning problems.

Original: Benchmarks.
Revised: Benchmarks.

Original: We evaluate our approach on nine NP -hard MILP problem benchmarks, which consist of three classical synthetic MILP problems and six challenging MILP problems from diverse application areas.
Revised: We evaluate our approach on nine NP -hard MILP problem benchmarks, which consist of three classical synthetic MILP problems and six challenging MILP problems from diverse application areas.

Original: We divide the nine problem benchmarks into three categories according to the difficulty of solving them using the SCIP 8.0.0 solver (Bestuzheva et al., 2021).
Revised: We divide the nine problem benchmarks into three categories according to the difficulty of solving them using the SCIP 8.0.0 solver (Bestuzheva et al., 2021).

Original: We call the three categories easy, medium, and hard datasets, respectively.
Revised: We call the three categories easy, medium, and hard datasets, respectively.

Original: (1) Easy datasets comprise three widely used synthetic MILP problem benchmarks: Set Covering (Balas & Ho, 1980), Maximum Independent Set (Bergman et al., 2016), and Multiple Knapsack (Scavuzzo et al., 2022).
Revised: (1) Easy datasets comprise three widely used synthetic MILP problem benchmarks: Set Covering (Balas & Ho, 1980), Maximum Independent Set (Bergman et al., 2016), and Multiple Knapsack (Scavuzzo et al., 2022).

Original: We artificially generate instances following Gasse et al.
Revised: We artificially generate instances following Gasse et al.

Original: ; Sun et al.
Revised: ; Sun et al.

Original: (2) Medium datasets comprise MIK (Atamt¨urk, 2003) and CORLAT (Gomes et al., 2008), which are widely used benchmarks for evaluating MILP solvers (He et al., 2014; Nair et al., 2020).
Revised: (2) Medium datasets comprise MIK (Atamt¨urk, 2003) and CORLAT (Gomes et al., 2008), which are widely used benchmarks for evaluating MILP solvers (He et al., 2014; Nair et al., 2020).

Original: (3) Hard datasets include the Load Balancing problem, inspired by real-life applications of large-scale systems, and the Anonymous problem, inspired by a large-scale industrial application (Bowly et al., 2021).
Revised: (3) Hard datasets include the Load Balancing problem, inspired by real-life applications of large-scale systems, and the Anonymous problem, inspired by a large-scale industrial application (Bowly et al., 2021).

Original: Moreover, hard datasets contain benchmarks from MIPLIB 2017 (MIPLIB) (Gleixner et al., 2021).
Revised: Moreover, hard datasets contain benchmarks from MIPLIB 2017 (MIPLIB) (Gleixner et al., 2021).

Original: Although Turner et al.
Revised: Although Turner et al.

Original: (2022) has shown that directly learning over the full MIPLIB can be extremely challenging, we propose to learn over subsets of MIPLIB.
Revised: (2022) has shown that directly learning over the full MIPLIB can be extremely challenging, we propose to learn over subsets of MIPLIB.

Original: We construct two subsets, called MIPLIB mixed neos and MIPLIB mixed supportcase.
Revised: We construct two subsets, called MIPLIB mixed neos and MIPLIB mixed supportcase.

Original: Due to limited space, please see Appendix D.1 for details of these datasets.
Revised: Due to limited space, please see Appendix D.1 for details of these datasets.

Original: Experimental setup.
Revised: Experimental setup.

Original: Throughout all experiments, we use SCIP 8.0.0 (Bestuzheva et al., 2021) as the backend solver, which is the state-of-the-art open source solver, and is widely used in research of machine learning for combinatorial optimization (Gasse et al., 2019;
Revised: Throughout all experiments, we use SCIP 8.0.0 (Bestuzheva et al., 2021) as the backend solver, which is the state-of-the-art open source solver, and is widely used in research of machine learning for combinatorial optimization (Gasse et al., 2019;

Original: Huang et al., 2022; Turner et al., 2022; Nair et al., 2020).
Revised: Huang et al., 2022; Turner et al., 2022; Nair et al., 2020).

Original: Following Gasse et al.
Revised: Following Gasse et al.

Original: ; Huang et al.
Revised: ; Huang et al.

Original: ; Paulus et al.
Revised: ; Paulus et al.

Original: , we only allow cutting plane generation and selection at the root node, and set the cut separation
Revised: , we only allow cutting plane generation and selection at the root node, and set the cut separation

Original: [Table]
Revised: [Table]

Original: rounds as one.
Revised: rounds as one.

Original: We keep all the other SCIP parameters to default so as to make comparisons as fair and reproducible as possible.
Revised: We keep all the other SCIP parameters to default so as to make comparisons as fair and reproducible as possible.

Original: We emphasize that all of the SCIP solver’s advanced features, such as presolve and heuristics, are open, which ensures that our setup is consistent with the practice setting.
Revised: We emphasize that all of the SCIP solver’s advanced features, such as presolve and heuristics, are open, which ensures that our setup is consistent with the practice setting.

Original: Throughout all experiments, we set the solving time limit as 300 seconds.
Revised: Throughout all experiments, we set the solving time limit as 300 seconds.

Original: For completeness, we also evaluate HEM with a much longer time limit of three hours.
Revised: For completeness, we also evaluate HEM with a much longer time limit of three hours.

Original: The results are given in
Revised: The results are given in

Original: Appendix G.6.
Revised: Appendix G.6.

Original: We train HEM with ADAM (Kingma & Ba, 2014) using the PyTorch (Paszke et al., 2019).
Revised: We train HEM with ADAM (Kingma & Ba, 2014) using the PyTorch (Paszke et al., 2019).

Original: Additionally, we also provide another implementation using the MindSpore (Chen, 2021).
Revised: Additionally, we also provide another implementation using the MindSpore (Chen, 2021).

Original: For simplicity, we split each dataset into the train and test sets with 80% and 20% instances.
Revised: For simplicity, we split each dataset into the train and test sets with 80% and 20% instances.

Original: To further improve HEM, one can construct a valid set for hyperparameters tuning.
Revised: To further improve HEM, one can construct a valid set for hyperparameters tuning.

Original: We train our model on the train set, and select the best model on the train set to evaluate on the test set.
Revised: We train our model on the train set, and select the best model on the train set to evaluate on the test set.

Original: Please refer to
Revised: Please refer to

Original: Appendix F.3 for implementation details, hyperparameters, and hardware specification.
Revised: Appendix F.3 for implementation details, hyperparameters, and hardware specification.

Original: Baselines.
Revised: Baselines.

Original: Our baselines include five widely used human-designed cut selection rules and a state-ofthe-art (SOTA) learning-based method.
Revised: Our baselines include five widely used human-designed cut selection rules and a state-ofthe-art (SOTA) learning-based method.

Original: Cut selection rules include NoCuts, Random, Normalized
Revised: Cut selection rules include NoCuts, Random, Normalized

Original: Violation (NV), Efficacy (Eff), and Default.
Revised: Violation (NV), Efficacy (Eff), and Default.

Original: NoCuts does not add any cuts.
Revised: NoCuts does not add any cuts.

Original: Default denotes the default cut selection rule used in SCIP 8.0.0.
Revised: Default denotes the default cut selection rule used in SCIP 8.0.0.

Original: For learning-based methods, we implement a slight variant of the SOTA learning-based methods (Tang et al., 2020; Huang et al., 2022), namely scorebased policy (SBP).
Revised: For learning-based methods, we implement a slight variant of the SOTA learning-based methods (Tang et al., 2020; Huang et al., 2022), namely scorebased policy (SBP).

Original: Please see Appendix F.2 for implementation details of these baselines.
Revised: Please see Appendix F.2 for implementation details of these baselines.

Original: Evaluation metrics.
Revised: Evaluation metrics.

Original: We use two widely used evaluation metrics, i.e., the average solving time (Time, lower is better), and the average primal-dual gap integral (PD integral, lower is better).
Revised: We use two widely used evaluation metrics, i.e., the average solving time (Time, lower is better), and the average primal-dual gap integral (PD integral, lower is better).

Original: Additionally, we provide more results in terms of another two metrics, i.e., the average number of nodes and the average primal-dual gap, in Appendix G.2.
Revised: Additionally, we provide more results in terms of another two metrics, i.e., the average number of nodes and the average primal-dual gap, in Appendix G.2.

Original: Furthermore, to evaluate different cut selection methods compared to pure branch-and-bound without cutting plane separation, we propose an Improvement metric.
Revised: Furthermore, to evaluate different cut selection methods compared to pure branch-and-bound without cutting plane separation, we propose an Improvement metric.

Original: Specifically, we define the metric by Im M ( · ) =
Revised: Specifically, we define the metric by Im M ( · ) =

Original: M ( NoCuts ) − M ( · ) M ( NoCuts ) , where
Revised: M ( NoCuts ) − M ( · ) M ( NoCuts ) , where

Original: M ( NoCuts ) represents the performance of NoCuts, and M ( · ) represents a mapping from a method to its performance.
Revised: M ( NoCuts ) represents the performance of NoCuts, and M ( · ) represents a mapping from a method to its performance.

Original: The improvement metric represents the improvement of a given method compared to NoCuts.
Revised: The improvement metric represents the improvement of a given method compared to NoCuts.

Original: We mainly focus on the Time metric on the easy datasets , as the solver can solve all instances to optimality within the given time limit.
Revised: We mainly focus on the Time metric on the easy datasets , as the solver can solve all instances to optimality within the given time limit.

Original: However, HEM and the baselines cannot solve all instances to optimality within the time limit on the medium and hard datasets.
Revised: However, HEM and the baselines cannot solve all instances to optimality within the time limit on the medium and hard datasets.

Original: As a result, the average solving time of those unsolved instances is the same, which makes it difficult to distinguish the performance of different cut selection methods using the Time metric.
Revised: As a result, the average solving time of those unsolved instances is the same, which makes it difficult to distinguish the performance of different cut selection methods using the Time metric.

Original: Therefore, we mainly focus on the PD integral metric on the medium and hard datasets .
Revised: Therefore, we mainly focus on the PD integral metric on the medium and hard datasets .

Original: The PD integral is also a well-recognized metric for evaluating the solver performance (Bowly et al., 2021; Cao et al., 2022).
Revised: The PD integral is also a well-recognized metric for evaluating the solver performance (Bowly et al., 2021; Cao et al., 2022).

Original: Experiment 1.
Revised: Experiment 1.

Original: Comparative evaluation
Revised: Comparative evaluation

Original: The results in Table 1 suggest the following.
Revised: The results in Table 1 suggest the following.

Original: (1) Easy datasets.
Revised: (1) Easy datasets.

Original: HEM significantly outperforms all the baselines on the easy datasets, especially on Max-
Revised: HEM significantly outperforms all the baselines on the easy datasets, especially on Max-

Original: [Table] [Table]
Revised: [Table] [Table]

Original: imum Independent Set and Multiple Knapsack.
Revised: imum Independent Set and Multiple Knapsack.

Original: SBP achieves much better performance than all the rule-based baselines, demonstrating that our implemented SBP is a strong baseline.
Revised: SBP achieves much better performance than all the rule-based baselines, demonstrating that our implemented SBP is a strong baseline.

Original: Compared to
Revised: Compared to

Original: SBP, HEM improves the Time by up to 16 .
Revised: SBP, HEM improves the Time by up to 16 .

Original: 4% on the three datasets, demonstrating the superiority of our method over the SOTA learning-based method.
Revised: 4% on the three datasets, demonstrating the superiority of our method over the SOTA learning-based method.

Original: (2) Medium datasets.
Revised: (2) Medium datasets.

Original: On MIK and CORLAT,
Revised: On MIK and CORLAT,

Original: HEM still outperforms all the baselines.
Revised: HEM still outperforms all the baselines.

Original: Especially on CORLAT, HEM achieves at least 33 .
Revised: Especially on CORLAT, HEM achieves at least 33 .

Original: 48% improvement in terms of the PD integral compared to the baselines.
Revised: 48% improvement in terms of the PD integral compared to the baselines.

Original: (3) Hard datasets.
Revised: (3) Hard datasets.

Original: HEM significantly outperforms the baselines in terms of the PD integral on several problems in the hard datasets.
Revised: HEM significantly outperforms the baselines in terms of the PD integral on several problems in the hard datasets.

Original: HEM achieves outstanding performance on two challenging datasets from MIPLIBand real-world problems (Load Balancing and Anonymous), demonstrating the powerful ability to enhance MILP solvers with HEM in large-scale real-world applications.
Revised: HEM achieves outstanding performance on two challenging datasets from MIPLIBand real-world problems (Load Balancing and Anonymous), demonstrating the powerful ability to enhance MILP solvers with HEM in large-scale real-world applications.

Original: Moreover, SBP performs extremely poorly on several medium and hard datasets, which implies that it can be difficult to learn good cut selection policies on challenging MILP problems.
Revised: Moreover, SBP performs extremely poorly on several medium and hard datasets, which implies that it can be difficult to learn good cut selection policies on challenging MILP problems.

Original: Experiment 2.
Revised: Experiment 2.

Original: Ablation study We present ablation studies on Maximum Independent Set (MIS),
Revised: Ablation study We present ablation studies on Maximum Independent Set (MIS),

Original: CORLAT, and MIPLIB mixed neos, which are representative datasets from the easy, medium, and hard datasets.
Revised: CORLAT, and MIPLIB mixed neos, which are representative datasets from the easy, medium, and hard datasets.

Original: We provide more results on the other datasets in Appendix G.3.
Revised: We provide more results on the other datasets in Appendix G.3.

Original: Contribution of each component.
Revised: Contribution of each component.

Original: We perform ablation studies to understand the contribution of each component in HEM.
Revised: We perform ablation studies to understand the contribution of each component in HEM.

Original: We report the performance of HEM and HEM without the higher-level model (HEM w/o H) in Table 2.
Revised: We report the performance of HEM and HEM without the higher-level model (HEM w/o H) in Table 2.

Original: HEM w/o H is essentially a pointer network.
Revised: HEM w/o H is essentially a pointer network.

Original: Note that it can still implicitly predicts the number of cuts that should be selected by predicting an end token as used in language tasks (Sutskever et al., 2014).
Revised: Note that it can still implicitly predicts the number of cuts that should be selected by predicting an end token as used in language tasks (Sutskever et al., 2014).

Original: Please see Appendix F.4.2 for details.
Revised: Please see Appendix F.4.2 for details.

Original: First , the results in
Revised: First , the results in

Original: Table 2 show that HEM w/o H outperforms all the baselines on MIS and CORLAT, demonstrating the advantages of the lower-level model.
Revised: Table 2 show that HEM w/o H outperforms all the baselines on MIS and CORLAT, demonstrating the advantages of the lower-level model.

Original: Although HEM w/o H outperforms Default on MIPLIB mixed neos, HEM w/o H performs on par with SBP.
Revised: Although HEM w/o H outperforms Default on MIPLIB mixed neos, HEM w/o H performs on par with SBP.

Original: A possible reason is that it is difficult for HEM w/o H to explore the action space efficiently, and thus HEM w/o H tends to be trapped to the local optimum.
Revised: A possible reason is that it is difficult for HEM w/o H to explore the action space efficiently, and thus HEM w/o H tends to be trapped to the local optimum.

Original: Second , the results in Table 2 show that HEM significantly outperforms HEM w/o H and the baselines on the three datasets.
Revised: Second , the results in Table 2 show that HEM significantly outperforms HEM w/o H and the baselines on the three datasets.

Original: The results demonstrate that the higher-level model is important for efficient exploration in complex tasks, thus significantly improving the solving efficiency.
Revised: The results demonstrate that the higher-level model is important for efficient exploration in complex tasks, thus significantly improving the solving efficiency.

Original: The importance of tackling (P1)-(P3).
Revised: The importance of tackling (P1)-(P3).

Original: We perform ablation studies to understand the importance of tackling (P1)-(P3) in cut selection.
Revised: We perform ablation studies to understand the importance of tackling (P1)-(P3) in cut selection.

Original: (1) HEM.
Revised: (1) HEM.

Original: HEM tackles (P1)-(P3) in cut selection simultaneously.
Revised: HEM tackles (P1)-(P3) in cut selection simultaneously.

Original: (2) HEM-ratio.
Revised: (2) HEM-ratio.

Original: In order not to learn how many cuts should be selected, we remove the higher-level model of HEM and force the lower-level model to select a fixed ratio of cuts .
Revised: In order not to learn how many cuts should be selected, we remove the higher-level model of HEM and force the lower-level model to select a fixed ratio of cuts .

Original: We denote it by HEM-ratio.
Revised: We denote it by HEM-ratio.

Original: Note that HEM-ratio is different from HEM w/o H (see Appendix F.4.2).
Revised: Note that HEM-ratio is different from HEM w/o H (see Appendix F.4.2).

Original: HEMratio tackles (P1) and (P3) in cut selection.
Revised: HEMratio tackles (P1) and (P3) in cut selection.

Original: (3) HEM-ratio-order.
Revised: (3) HEM-ratio-order.

Original: To further mute the effect of the order of selected cuts, we reorder the selected cuts given by HEM-ratio with the original index of the generated cuts, which we denote by HEM-ratio-order.
Revised: To further mute the effect of the order of selected cuts, we reorder the selected cuts given by HEM-ratio with the original index of the generated cuts, which we denote by HEM-ratio-order.

Original: HEM-ratio-order mainly tackles (P1) in cut selection.
Revised: HEM-ratio-order mainly tackles (P1) in cut selection.

Original: The results in Table 3 suggest the following.
Revised: The results in Table 3 suggest the following.

Original: HEM-ratio-order significantly outperforms
Revised: HEM-ratio-order significantly outperforms

Original: Default and NoCuts, demonstrating that tackling (P1) by data-driven methods is crucial.
Revised: Default and NoCuts, demonstrating that tackling (P1) by data-driven methods is crucial.

Original: HEM significantly outperforms HEM-ratio in terms of the PD integral, demonstrating the significance of tackling (P2) .
Revised: HEM significantly outperforms HEM-ratio in terms of the PD integral, demonstrating the significance of tackling (P2) .

Original: HEM-ratio outperforms HEM-ratio-order in terms of the Time and the PD integral, which demonstrates the importance of tackling (P3) .
Revised: HEM-ratio outperforms HEM-ratio-order in terms of the Time and the PD integral, which demonstrates the importance of tackling (P3) .

Original: Moreover, HEM-ratio and HEM-ratio-order perform better than SBP on MIS and CORLAT, demonstrating the advantages of using the sequence model to learn cut selection over SBP.
Revised: Moreover, HEM-ratio and HEM-ratio-order perform better than SBP on MIS and CORLAT, demonstrating the advantages of using the sequence model to learn cut selection over SBP.

Original: HEM-ratio and HEM-ratio-order perform on par with SBP on MIPLIB mixed neos.
Revised: HEM-ratio and HEM-ratio-order perform on par with SBP on MIPLIB mixed neos.

Original: We provide possible reasons in Appendix G.3.1.
Revised: We provide possible reasons in Appendix G.3.1.

Original: [Table]
Revised: [Table]

Original: [Figure]
Revised: [Figure]

Original: Experiment 3.
Revised: Experiment 3.

Original: Generalization We evaluate the ability of HEM to generalize across different sizes of MILPs.
Revised: Generalization We evaluate the ability of HEM to generalize across different sizes of MILPs.

Original: Let n × m denote the size of MILP instances.
Revised: Let n × m denote the size of MILP instances.

Original: Following Gasse et al.
Revised: Following Gasse et al.

Original: ; Sun et al.
Revised: ; Sun et al.

Original: (2020), we test the generalization ability of HEM on Set Covering and Maximum Independent Set (MIS), as we can artificially generate instances with arbitrary sizes for synthetic MILP problems.
Revised: (2020), we test the generalization ability of HEM on Set Covering and Maximum Independent Set (MIS), as we can artificially generate instances with arbitrary sizes for synthetic MILP problems.

Original: On MIS, we test HEM on four times and nine times larger instances than those seen during training.
Revised: On MIS, we test HEM on four times and nine times larger instances than those seen during training.

Original: The results in Table 4 ( Left ) show that HEM significantly outperforms the baselines in terms of the
Revised: The results in Table 4 ( Left ) show that HEM significantly outperforms the baselines in terms of the

Original: Time and the PD integral on 4 × and 9 × MIS, demonstrating the superiority of HEM in terms of the generalization ability.
Revised: Time and the PD integral on 4 × and 9 × MIS, demonstrating the superiority of HEM in terms of the generalization ability.

Original: Interestingly, SBP also generalizes well to large instances, demonstrating that
Revised: Interestingly, SBP also generalizes well to large instances, demonstrating that

Original: SBP is a strong baseline.
Revised: SBP is a strong baseline.

Original: We provide more results on Set Covering in Appendix G.4.
Revised: We provide more results on Set Covering in Appendix G.4.

Original: [Figure]
Revised: [Figure]

Original: Colored points illustrate the reduced cut features.
Revised: Colored points illustrate the reduced cut features.

Original: The area covered by the dashed lines represents the diversity of selected cuts.
Revised: The area covered by the dashed lines represents the diversity of selected cuts.

Original: The results show that
Revised: The results show that

Original: HEM-ratio selects much more diverse cuts than SBP.
Revised: HEM-ratio selects much more diverse cuts than SBP.

Original: Experiment 4.
Revised: Experiment 4.

Original: Visualization of selected cuts We visualize the diversity of selected cuts, an important metric for evaluating whether the selected cuts complement each other nicely (Dey & Molinaro, 2018b).
Revised: Visualization of selected cuts We visualize the diversity of selected cuts, an important metric for evaluating whether the selected cuts complement each other nicely (Dey & Molinaro, 2018b).

Original: We visualize the cuts selected by HEM-ratio and SBP on a randomly sampled instance from Maximum Independent Set and CORLAT, respectively.
Revised: We visualize the cuts selected by HEM-ratio and SBP on a randomly sampled instance from Maximum Independent Set and CORLAT, respectively.

Original: We evaluate HEM-ratio rather than
Revised: We evaluate HEM-ratio rather than

Original: HEM, as HEM-ratio selects the same number of cuts as SBP.
Revised: HEM, as HEM-ratio selects the same number of cuts as SBP.

Original: Furthermore, we perform principal component analysis on the selected cuts to reduce the cut features to two-dimensional space.
Revised: Furthermore, we perform principal component analysis on the selected cuts to reduce the cut features to two-dimensional space.

Original: Colored points illustrate the reduced cut features.
Revised: Colored points illustrate the reduced cut features.

Original: To visualize the diversity of selected cuts, we use dashed lines to connect the points with the smallest and largest x,y coordinates.
Revised: To visualize the diversity of selected cuts, we use dashed lines to connect the points with the smallest and largest x,y coordinates.

Original: That is, the area covered by the dashed lines represents the diversity .
Revised: That is, the area covered by the dashed lines represents the diversity .

Original: Figure 3 shows that SBP tends to select many similar cuts that are possibly redundant, especially on Maximum Independent Set.
Revised: Figure 3 shows that SBP tends to select many similar cuts that are possibly redundant, especially on Maximum Independent Set.

Original: In contrast, HEM-ratio selects much more diverse cuts that can well complement each other.
Revised: In contrast, HEM-ratio selects much more diverse cuts that can well complement each other.

Original: Please refer to Appendix G.5 for results on more datasets.
Revised: Please refer to Appendix G.5 for results on more datasets.

Original: Experiment 5.
Revised: Experiment 5.

Original: Deployment in real-world production planning problems To further evaluate the effectiveness of our proposed HEM, we deploy HEM to large-scale real-world production planning problems 2 at an anonymous enterprise, which is one of the largest global commercial technology enterprises.
Revised: Deployment in real-world production planning problems To further evaluate the effectiveness of our proposed HEM, we deploy HEM to large-scale real-world production planning problems 2 at an anonymous enterprise, which is one of the largest global commercial technology enterprises.

Original: Please refer to Appendix D.3 for more details of the problems.
Revised: Please refer to Appendix D.3 for more details of the problems.

Original: The results in Table( Right ) show that HEM significantly outperforms all the baselines in terms of the Time and PD integral.
Revised: The results in Table( Right ) show that HEM significantly outperforms all the baselines in terms of the Time and PD integral.

Original: The results demonstrate the strong ability to enhance modern MILP solvers with our proposed
Revised: The results demonstrate the strong ability to enhance modern MILP solvers with our proposed

Original: HEM in real-world applications.
Revised: HEM in real-world applications.

Original: Interestingly, Default performs poorer than NoCuts, which implies that an improper cut selection policy could significantly degrade the performance of MILP solvers.
Revised: Interestingly, Default performs poorer than NoCuts, which implies that an improper cut selection policy could significantly degrade the performance of MILP solvers.

Original:  C ONCLUSION
Revised:  C ONCLUSION

Original: In this paper, we observe from extensive empirical results that the order of selected cuts has a significant impact on the efficiency of solving MILPs.
Revised: In this paper, we observe from extensive empirical results that the order of selected cuts has a significant impact on the efficiency of solving MILPs.

Original: We propose a novel h ierarchical s e quence m odel
Revised: We propose a novel h ierarchical s e quence m odel

Original: (HEM) to learn cut selection policies via reinforcement learning.
Revised: (HEM) to learn cut selection policies via reinforcement learning.

Original: Specifically, HEM consists of a two-level model: (1) a higher-level model to learn the number of cuts that should be selected, (2) and a lower-level model—that formulates the cut selection task as a sequence to sequence learning problem—to learn policies selecting an ordered subset with the size determined by the higher-level model.
Revised: Specifically, HEM consists of a two-level model: (1) a higher-level model to learn the number of cuts that should be selected, (2) and a lower-level model—that formulates the cut selection task as a sequence to sequence learning problem—to learn policies selecting an ordered subset with the size determined by the higher-level model.

Original: Experiments show that HEM significantly improves the efficiency of solving MILPs compared to human-designed and learning-based baselines on both synthetic and large-scale real-world
Revised: Experiments show that HEM significantly improves the efficiency of solving MILPs compared to human-designed and learning-based baselines on both synthetic and large-scale real-world

Original: MILPs.
Revised: MILPs.

Original: We believe that our proposed approach brings new insights into learning cut selection.
Revised: We believe that our proposed approach brings new insights into learning cut selection.

Original:  E THICS S TATEMENT
Revised:  E THICS S TATEMENT

Original: As our work is fundamental research in solving mixed-integer linear programs (MILPs) there are no direct ethical risks.
Revised: As our work is fundamental research in solving mixed-integer linear programs (MILPs) there are no direct ethical risks.

Original: Modern MILP solvers with expert-designed cut selection heuristics aim to solve general MILPs, while modern MILP solvers with our proposed HEM can learn more effective cut selection heuristics from MILPs collected from certain types of real-world applications
Revised: Modern MILP solvers with expert-designed cut selection heuristics aim to solve general MILPs, while modern MILP solvers with our proposed HEM can learn more effective cut selection heuristics from MILPs collected from certain types of real-world applications.

Original:  R EPRODUCIBILITY S TATEMENT
Revised:  R EPRODUCIBILITY S TATEMENT

Original: To ensure reproducibility we describe our proposed HEM in Section 4 and provide more implementation details in Appendix F. Moreover, we provide the pseudocode of our proposed HEM in
Revised: To ensure reproducibility we describe our proposed HEM in Section 4 and provide more implementation details in Appendix F. Moreover, we provide the pseudocode of our proposed HEM in

Original: Appendix F.3.6.
Revised: Appendix F.3.6.

Original: We will release all the source codes publicly once the paper is accepted.
Revised: We will release all the source codes publicly once the paper is accepted.

Original: All datasets except the real-world production planning problems used in the experiments are publicly available.
Revised: All datasets except the real-world production planning problems used in the experiments are publicly available.

Original: Please refer to Appendix D for details of these datasets.
Revised: Please refer to Appendix D for details of these datasets.

Original: We will also publicly release the dataset from the real-world production planning problems once the paper is accepted.
Revised: We will also publicly release the dataset from the real-world production planning problems once the paper is accepted.

Original: A P ROOF A. P ROOF OF P ROPOSITION
Revised: A P ROOF A. P ROOF OF P ROPOSITION

Original: Proof.
Revised: Proof.

Original: The optimization objective takes the form of
Revised: The optimization objective takes the form of

Original: [Equation]
Revised: [Equation]

Original: where θ = [ θ
Revised: where θ = [ θ

Original: [Equation]
Revised: [Equation]

Original: θ 2 ] with [ θ
Revised: θ 2 ] with [ θ

Original: [Equation]
Revised: [Equation]

Original: θ 2 ] denoting the concatenation of the two vectors
Revised: θ 2 ] denoting the concatenation of the two vectors

Original: [Equation]
Revised: [Equation]

Original: π θ ( a k | s
Revised: π θ ( a k | s

Original: [Equation]
Revised: [Equation]

Original: =
Revised: =

Original: E k ∼ π hθ
Revised: E k ∼ π hθ

Original: [Equation]
Revised: [Equation]

Original: ·| s ) [ π hθ 2 ( a k | s, k )]
Revised: ·| s ) [ π hθ 2 ( a k | s, k )]

Original: [Equation]
Revised: [Equation]

Original: and µ denotes the initial state distribution.
Revised: and µ denotes the initial state distribution.

Original: We first compute the policy gradient for θ 1 :
Revised: We first compute the policy gradient for θ 1 :

Original: [Equation]
Revised: [Equation]

Original: Let
Revised: Let

Original: [Equation]
Revised: [Equation]

Original: then we have that
Revised: then we have that

Original: [Equation]
Revised: [Equation]

Original: Therefore
Revised: Therefore

Original: [Equation]
Revised: [Equation]

Original: we have that
Revised: we have that

Original: [Equation]
Revised: [Equation]

Original: We then compute the policy gradient for θ 2 :
Revised: We then compute the policy gradient for θ 2 :

Original: [Equation]
Revised: [Equation]

Original: which completes the proof
Revised: which completes the proof

Original: [Equation]
Revised: [Equation]

Original: B R ELATED W ORK
Revised: B R ELATED W ORK

Original: Machine learning for MILP.
Revised: Machine learning for MILP.

Original: The use of machine learning methods to help improve the MILP solver performance has been an active topic of significant interest in recent years (Bengio et al.,
Revised: The use of machine learning methods to help improve the MILP solver performance has been an active topic of significant interest in recent years (Bengio et al.,

Original: ; Lodi & Zarpellon, 2017; Bowly et al., 2021; Gasse et al., 2019).
Revised: ; Lodi & Zarpellon, 2017; Bowly et al., 2021; Gasse et al., 2019).

Original: During the solving process of the solvers, many crucial decisions that significantly impact the solver performance are based on heuristics (Achterberg, 2007).
Revised: During the solving process of the solvers, many crucial decisions that significantly impact the solver performance are based on heuristics (Achterberg, 2007).

Original: Recent methods propose to replace these hand-crafted heuristics with machine learning models (Bengio et al., 2021).
Revised: Recent methods propose to replace these hand-crafted heuristics with machine learning models (Bengio et al., 2021).

Original: This line of research has shown significant improvement on the solver performance, including cut selection (Tang et al., 2020; Paulus et al., 2022; Turner et al., 2022; Baltean-Lugojan et al., 2019), variable selection (Khalil et al., 2016; Gasse et al., 2019; Balcan et al., 2018; Zarpellon et al., 2021), node selection (
Revised: This line of research has shown significant improvement on the solver performance, including cut selection (Tang et al., 2020; Paulus et al., 2022; Turner et al., 2022; Baltean-Lugojan et al., 2019), variable selection (Khalil et al., 2016; Gasse et al., 2019; Balcan et al., 2018; Zarpellon et al., 2021), node selection (

Original: He et al., 2014; Sabharwal et al., 2012), column generation (Morabit et al., 2021), and primal heuristics selection (Khalil et al., 2017; Hendel et al., 2019).
Revised: He et al., 2014; Sabharwal et al., 2012), column generation (Morabit et al., 2021), and primal heuristics selection (Khalil et al., 2017; Hendel et al., 2019).

Original: In this paper, we focus on cut selection, which plays a significant role in modern MILP solvers (Dey & Molinaro, 2018a; Tang et al., 2020).
Revised: In this paper, we focus on cut selection, which plays a significant role in modern MILP solvers (Dey & Molinaro, 2018a; Tang et al., 2020).

Original: For cut selection, many existing learning-based methods (Tang et al., 2020; Paulus et al., 2022;
Revised: For cut selection, many existing learning-based methods (Tang et al., 2020; Paulus et al., 2022;

Original: Huang et al., 2022) focus on learning which cuts should be preferred by learning a scoring function to measure cut quality.
Revised: Huang et al., 2022) focus on learning which cuts should be preferred by learning a scoring function to measure cut quality.

Original: Specifically, (Tang et al., 2020) proposes a reinforcement learning approach to learn to score Gomory cuts (Gomory, 1960) and select a Gomory cut with the best scores.
Revised: Specifically, (Tang et al., 2020) proposes a reinforcement learning approach to learn to score Gomory cuts (Gomory, 1960) and select a Gomory cut with the best scores.

Original: Furthermore, (Paulus et al., 2022) designs a lookahead selection rule which selects a cut that yields the best dual bound improvement, and proposes to learn the expert rule via imitation learning.
Revised: Furthermore, (Paulus et al., 2022) designs a lookahead selection rule which selects a cut that yields the best dual bound improvement, and proposes to learn the expert rule via imitation learning.

Original: Instead of selecting the best cut, (Huang et al., 2022) frames cut selection as multiple instance learning to learn a scoring function and selects a fixed ratio of cuts with high scores.
Revised: Instead of selecting the best cut, (Huang et al., 2022) frames cut selection as multiple instance learning to learn a scoring function and selects a fixed ratio of cuts with high scores.

Original: However, they neglect the importance of learning how many cuts should be selected.
Revised: However, they neglect the importance of learning how many cuts should be selected.

Original: Moreover, we empirically show that the order of selected cuts has a large impact on the efficiency of solving MILPs (see Section 3).
Revised: Moreover, we empirically show that the order of selected cuts has a large impact on the efficiency of solving MILPs (see Section 3).

Original: Moreover, (Turner et al., 2022) proposes to learn the weightings of four existing scoring rules designed by experts.
Revised: Moreover, (Turner et al., 2022) proposes to learn the weightings of four existing scoring rules designed by experts.

Original: For the theoretical analysis, (Balcan et al., 2021) provides some provable guarantees for learning cut selection policies.
Revised: For the theoretical analysis, (Balcan et al., 2021) provides some provable guarantees for learning cut selection policies.

Original: Sequence model.
Revised: Sequence model.

Original: Sequence model such as long-short term memory and Transformer has achieved outstanding performance in language tasks such as machine translation (Hochreiter & Schmidhuber, 1997; Sutskever et al., 2014; Vaswani et al., 2017).
Revised: Sequence model such as long-short term memory and Transformer has achieved outstanding performance in language tasks such as machine translation (Hochreiter & Schmidhuber, 1997; Sutskever et al., 2014; Vaswani et al., 2017).

Original: For combinatorial optimization, recent works (Vinyals et al., 2015; Bello* et al., 2017) propose a variant of the traditional sequence model, namely pointer network, which is applied to directly finding solutions for specific combinatorial optimization problems, such as the Travelling Salesman Problem (Lenstra & Shmoys, 2009).
Revised: For combinatorial optimization, recent works (Vinyals et al., 2015; Bello* et al., 2017) propose a variant of the traditional sequence model, namely pointer network, which is applied to directly finding solutions for specific combinatorial optimization problems, such as the Travelling Salesman Problem (Lenstra & Shmoys, 2009).

Original: Instead of finding solutions directly, we propose to use the pointer network for cut selection in modern MILP solvers.
Revised: Instead of finding solutions directly, we propose to use the pointer network for cut selection in modern MILP solvers.

Original: To the best of our knowledge, we are the first to apply the pointer network to cut selection, which not only captures the order of selected cuts, but also can well capture the interaction among cuts to select cuts that complement each other nicely.
Revised: To the best of our knowledge, we are the first to apply the pointer network to cut selection, which not only captures the order of selected cuts, but also can well capture the interaction among cuts to select cuts that complement each other nicely.

Original: C M ORE D ETAILS OF B ACKGROUND C. M ORE DETAILS OF THE PRIMAL - DUAL GAP INTEGRAL
Revised: C M ORE D ETAILS OF B ACKGROUND C. M ORE DETAILS OF THE PRIMAL - DUAL GAP INTEGRAL

Original: We keep track of two important bounds when running branch-and-cut, including the global primal and dual bound.
Revised: We keep track of two important bounds when running branch-and-cut, including the global primal and dual bound.

Original: The global primal bound corresponds to the value of the best feasible solution found so far, which is the best upper bound of the problem in (1).
Revised: The global primal bound corresponds to the value of the best feasible solution found so far, which is the best upper bound of the problem in (1).

Original: The global dual bound corresponds to the minimum dual bound across all leaves of the search tree, which is the best lower bound of the problem in (1).
Revised: The global dual bound corresponds to the minimum dual bound across all leaves of the search tree, which is the best lower bound of the problem in (1).

Original: We define the primal-dual gap integral by the area between the curve of the solver’s global primal bound and the curve of the solver’s global dual bound.
Revised: We define the primal-dual gap integral by the area between the curve of the solver’s global primal bound and the curve of the solver’s global dual bound.

Original: With a time limit T , we define the primal-dual gap integral by
Revised: With a time limit T , we define the primal-dual gap integral by

Original: [Equation]
Revised: [Equation]

Original: where c is the objective coefficient vector as in (1), x ∗ t is the best feasible solution found at time t , z ∗ t is the best dual bound at time t .
Revised: where c is the objective coefficient vector as in (1), x ∗ t is the best feasible solution found at time t , z ∗ t is the best dual bound at time t .

Original: We define the primal-dual gap by the difference between the global primal bound and the global dual bound.
Revised: We define the primal-dual gap by the difference between the global primal bound and the global dual bound.

Original: In SCIP 8.0.0 (Bestuzheva et al., 2021), the initial value of the primal-dual gap is set to a constant 100.
Revised: In SCIP 8.0.0 (Bestuzheva et al., 2021), the initial value of the primal-dual gap is set to a constant 100.

Original: The primal-dual gap integral is a well-recognized metric for evaluating solver performance.
Revised: The primal-dual gap integral is a well-recognized metric for evaluating solver performance.

Original: For example, the primal-dual gal integral is a primary evaluation metric in the NeurIPS 2021 ML4CO competition (Bowly et al., 2021).
Revised: For example, the primal-dual gal integral is a primary evaluation metric in the NeurIPS 2021 ML4CO competition (Bowly et al., 2021).

Original: D D ETAILS OF THE D ATASETS U SED IN THIS P APER D. T HE DATASETS USED IN THE MAIN EVALUATION
Revised: D D ETAILS OF THE D ATASETS U SED IN THIS P APER D. T HE DATASETS USED IN THE MAIN EVALUATION

Original: Easy datasets.
Revised: Easy datasets.

Original: The SCIP 8.0.0 solver needs one minute to solve the MILP instances in the easy datasets to optimality.
Revised: The SCIP 8.0.0 solver needs one minute to solve the MILP instances in the easy datasets to optimality.

Original: Easy datasets are comprised of three synthetic MILP problems: Set Covering (Balas & Ho, 1980), Maximum Independent Set (Bergman et al., 2016), and Multiple Knapsack (Scavuzzo et al., 2022).
Revised: Easy datasets are comprised of three synthetic MILP problems: Set Covering (Balas & Ho, 1980), Maximum Independent Set (Bergman et al., 2016), and Multiple Knapsack (Scavuzzo et al., 2022).

Original: We choose these three classes of problems for the following reasons.
Revised: We choose these three classes of problems for the following reasons.

Original: First, they are widely used benchmarks for evaluating MILP solvers (Gasse et al., 2019;
Revised: First, they are widely used benchmarks for evaluating MILP solvers (Gasse et al., 2019;

Original: Huang et al., 2022; Sun et al., 2020; Gupta et al., 2022).
Revised: Huang et al., 2022; Sun et al., 2020; Gupta et al., 2022).

Original: Second, they represent a wide collection of MILP problems encountered in practice.
Revised: Second, they represent a wide collection of MILP problems encountered in practice.

Original: Third, for each class of these problems, the average number of generated cuts is at least twenty, which ensures that proper cut selection strategies are significant for improving the solver performance.
Revised: Third, for each class of these problems, the average number of generated cuts is at least twenty, which ensures that proper cut selection strategies are significant for improving the solver performance.

Original: Similarly to (Gasse et al., 2019; Scavuzzo et al., 2022; Sun et al., 2020; Gupta et al., 2022), we generate set covering instances with 500 rows and 1000 columns, Maximum Independent Set instances with graphs of 500 nodes and affinity set to 4, multiple knapsack instances with 60 items and 12 knapsacks.
Revised: Similarly to (Gasse et al., 2019; Scavuzzo et al., 2022; Sun et al., 2020; Gupta et al., 2022), we generate set covering instances with 500 rows and 1000 columns, Maximum Independent Set instances with graphs of 500 nodes and affinity set to 4, multiple knapsack instances with 60 items and 12 knapsacks.

Original: For each benchmark, we generate a training set of 10,000 instances, and a test set of 100 instances that are never seen during training.
Revised: For each benchmark, we generate a training set of 10,000 instances, and a test set of 100 instances that are never seen during training.

Original: Specifically, readers can refer to https://github.com/ds4dm/learn2branch or https://github.com/lascavana/rl2branch for code to generate the easy datasets.
Revised: Specifically, readers can refer to https://github.com/ds4dm/learn2branch or https://github.com/lascavana/rl2branch for code to generate the easy datasets.

Original: We will also release our code once the paper is accepted to be published.
Revised: We will also release our code once the paper is accepted to be published.

Original: Medium datasets.
Revised: Medium datasets.

Original: The SCIP 8.0.0 solver needs at least five minutes to solve the instances in the medium datasets to optimality.
Revised: The SCIP 8.0.0 solver needs at least five minutes to solve the instances in the medium datasets to optimality.

Original: Following He et al.
Revised: Following He et al.

Original: ; Hutter et al.
Revised: ; Hutter et al.

Original: ; Nair et al.
Revised: ; Nair et al.

Original: (2020), medium datasets comprise MIK (Atamt ¨ urk, 2003), a set of MILP problems with knapsack constraints, and CORLAT (Gomes et al., 2008), a real dataset used for the construction of a wildlife corridor for grizzly bears in the Northern Rockies region.
Revised: (2020), medium datasets comprise MIK (Atamt ¨ urk, 2003), a set of MILP problems with knapsack constraints, and CORLAT (Gomes et al., 2008), a real dataset used for the construction of a wildlife corridor for grizzly bears in the Northern Rockies region.

Original: Each problem set is split into training and test sets with 80% and 20% of the instances.
Revised: Each problem set is split into training and test sets with 80% and 20% of the instances.

Original: Readers can refer to https://atamturk.
Revised: Readers can refer to https://atamturk.

Original: ieor.berkeley.edu/data/mixed.integer.knapsack/ for MIK.
Revised: ieor.berkeley.edu/data/mixed.integer.knapsack/ for MIK.

Original: Readers can refer to https://bitbucket.org/mlindauer/aclib2/src/master/ for CORLAT.
Revised: Readers can refer to https://bitbucket.org/mlindauer/aclib2/src/master/ for CORLAT.

Original: [Table]
Revised: [Table]

Original: Hard datasets.
Revised: Hard datasets.

Original: The SCIP 8.0.
Revised: The SCIP 8.0.

Original: solver needs at least one hour to solve the instances in the hard datasets to optimality.
Revised: solver needs at least one hour to solve the instances in the hard datasets to optimality.

Original: (1) Benchmarks from MIPLIB 2017.
Revised: (1) Benchmarks from MIPLIB 2017.

Original: Note that MIPLIB 2017 (MIPLIB) (Gleixner et al., 2021) contains instances of MILPs across many different application areas and has been used as a long-standing standard benchmark for MILP solvers (Nair et al., 2020; Turner et al., 2022; Gleixner et al., 2021).
Revised: Note that MIPLIB 2017 (MIPLIB) (Gleixner et al., 2021) contains instances of MILPs across many different application areas and has been used as a long-standing standard benchmark for MILP solvers (Nair et al., 2020; Turner et al., 2022; Gleixner et al., 2021).

Original: Previous work (Turner et al., 2022) has shown that directly learning over the full
Revised: Previous work (Turner et al., 2022) has shown that directly learning over the full

Original: MIPLIB can be extremely challenging, as these instances are heterogeneous but machine learning
Revised: MIPLIB can be extremely challenging, as these instances are heterogeneous but machine learning

Original: [Table]
Revised: [Table]

Original: has difficulty in learning from heterogeneous datasets.
Revised: has difficulty in learning from heterogeneous datasets.

Original: Despite this challenge, we take the first step towards learning over subsets of MIPLIB.
Revised: Despite this challenge, we take the first step towards learning over subsets of MIPLIB.

Original: Specifically, we construct two subsets by selecting similar instances from MIPLIB.
Revised: Specifically, we construct two subsets by selecting similar instances from MIPLIB.

Original: We measure the similarity between instances by 100 human-designed instance features (Gleixner et al., 2021).
Revised: We measure the similarity between instances by 100 human-designed instance features (Gleixner et al., 2021).

Original: Following Turner et al.
Revised: Following Turner et al.

Original: , we first discard instances from MILLIB that satisfy any of the criteria in Table 5.
Revised: , we first discard instances from MILLIB that satisfy any of the criteria in Table 5.

Original: This ensures that a good cut selection policy can significantly improve the dual bound on the remaining instances.
Revised: This ensures that a good cut selection policy can significantly improve the dual bound on the remaining instances.

Original: Note that we only use three of seven criteria that are used in (Turner et al., 2022) to preserve as many instances as possible.
Revised: Note that we only use three of seven criteria that are used in (Turner et al., 2022) to preserve as many instances as possible.

Original: To select similar instances from MIPLIB 2017, we first choose a representative instance with knapsack constraints (neos-1456979), and a representative instance with set covering constraints (supportcase40).
Revised: To select similar instances from MIPLIB 2017, we first choose a representative instance with knapsack constraints (neos-1456979), and a representative instance with set covering constraints (supportcase40).

Original: Then we construct the dataset MIPLIB mixed neos following the procedure in Algorithm 1 with the initial instance neos-1456979.
Revised: Then we construct the dataset MIPLIB mixed neos following the procedure in Algorithm 1 with the initial instance neos-1456979.

Original: We construct the dataset MIPLIB mixed supportcase following the procedure in Algorithm 1 with the initial instance supportcase40.
Revised: We construct the dataset MIPLIB mixed supportcase following the procedure in Algorithm 1 with the initial instance supportcase40.

Original: Note that We measure the similarity between instances by 100 human-designed instance features (Gleixner et al., 2021).
Revised: Note that We measure the similarity between instances by 100 human-designed instance features (Gleixner et al., 2021).

Original: Each dataset is split into training and test sets with 80% and 20% of the instances.
Revised: Each dataset is split into training and test sets with 80% and 20% of the instances.

Original: Specifically, MIPLIB mixed neos contains 20 instances: neos-1456979, ic97 tension, icir97 tension, l2p12, lectsched-4-obj, lectsched-5-obj, loopha13, neos-686190, neos-2294525-abba, neos3009394-lami, neos-3046601-motu, neos-3046615-murg, neos-3610173-itata, neos-4338804snowy, neos-5221106-oparau, neos-5260764-orauea, neos-5261882-treska, neos-5266653-tugela, neos16, and timtab1CUTS.
Revised: Specifically, MIPLIB mixed neos contains 20 instances: neos-1456979, ic97 tension, icir97 tension, l2p12, lectsched-4-obj, lectsched-5-obj, loopha13, neos-686190, neos-2294525-abba, neos3009394-lami, neos-3046601-motu, neos-3046615-murg, neos-3610173-itata, neos-4338804snowy, neos-5221106-oparau, neos-5260764-orauea, neos-5261882-treska, neos-5266653-tugela, neos16, and timtab1CUTS.

Original: Moreover, MIPLIB mixed supportcase contains 40 instances: supportcase40, 30 70 45 05 100, 30 70 45 095 100, acc-tight2, acc-tight4, acc-tight5, comp07-2idx, comp08-2idx, comp122idx, comp21-2idx, decomp1, decomp2, gus-sch, istanbul-no-cutoff, mkc, mkc1, neos-555343, neos-555424, neos-738098, neos-872648, neos-933562, neos-933638, neos-933966, neos935234, neos-935769, neos-983171, neos-1330346, neos-1337307, neos-1396125, neos-3209462rhin, neos-3755335-nizao, neos-3759587-noosa, neos-4300652-rahue, neos18, physiciansched6-1, physiciansched6-2, piperout-d27, qiu, reblock354, and supportcase37.
Revised: Moreover, MIPLIB mixed supportcase contains 40 instances: supportcase40, 30 70 45 05 100, 30 70 45 095 100, acc-tight2, acc-tight4, acc-tight5, comp07-2idx, comp08-2idx, comp122idx, comp21-2idx, decomp1, decomp2, gus-sch, istanbul-no-cutoff, mkc, mkc1, neos-555343, neos-555424, neos-738098, neos-872648, neos-933562, neos-933638, neos-933966, neos935234, neos-935769, neos-983171, neos-1330346, neos-1337307, neos-1396125, neos-3209462rhin, neos-3755335-nizao, neos-3759587-noosa, neos-4300652-rahue, neos18, physiciansched6-1, physiciansched6-2, piperout-d27, qiu, reblock354, and supportcase37.

Original: (2) Benchmarks used in NeurIPS 2021 ML4CO competition The Load Balancing and Anonymous problems used in the main text are from the NeurIPS 2021 ML4CO competition (Bowly et al., 2021).
Revised: (2) Benchmarks used in NeurIPS 2021 ML4CO competition The Load Balancing and Anonymous problems used in the main text are from the NeurIPS 2021 ML4CO competition (Bowly et al., 2021).

Original: Readers can refer to https://www.ecole.ai/2021/ml4co-competition/ for details of the competition.
Revised: Readers can refer to https://www.ecole.ai/2021/ml4co-competition/ for details of the competition.

Original: The competition releases three challenging datasets, but we only use two of the three datasets.
Revised: The competition releases three challenging datasets, but we only use two of the three datasets.

Original: The major reason is that the average number of the candidate cuts on the instances from the third dataset (Item Placement) is less than five, which makes cut selection has little impact on the overall solver performance.
Revised: The major reason is that the average number of the candidate cuts on the instances from the third dataset (Item Placement) is less than five, which makes cut selection has little impact on the overall solver performance.

Original: D.1. D ETAILED DESCRIPTION OF THE AFOREMENTIONED DATASETS
Revised: D.1. D ETAILED DESCRIPTION OF THE AFOREMENTIONED DATASETS

Original: In this part, we provide detailed description of the aforementioned datasets.
Revised: In this part, we provide detailed description of the aforementioned datasets.

Original: Note that all datasets we use except MIPLIB 2017 are application-specific, i.e., they contain instances from only a single application.
Revised: Note that all datasets we use except MIPLIB 2017 are application-specific, i.e., they contain instances from only a single application.

Original: We summarize the statistical description of the used datasets in this paper in Table 6.
Revised: We summarize the statistical description of the used datasets in this paper in Table 6.

Original: Let n, m denote the average number of variables and constraints in the MILPs.
Revised: Let n, m denote the average number of variables and constraints in the MILPs.

Original: Let m × n denote the size of the MILPs.
Revised: Let m × n denote the size of the MILPs.

Original: We emphasize that the largest size of our used datasets is up to two orders of magnitude larger than that used in previous learning-based cut selection methods (Tang et al., 2020;
Revised: We emphasize that the largest size of our used datasets is up to two orders of magnitude larger than that used in previous learning-based cut selection methods (Tang et al., 2020;

Original: Paulus et al., 2022), which demonstrates the superiority of our proposed HEM.
Revised: Paulus et al., 2022), which demonstrates the superiority of our proposed HEM.

Original: Moreover, we test the inference time of our proposed HEM given the average number of candidate cuts.
Revised: Moreover, we test the inference time of our proposed HEM given the average number of candidate cuts.

Original: The results in
Revised: The results in

Original: Table 6 show that the computational overhead of the HEM is very low.
Revised: Table 6 show that the computational overhead of the HEM is very low.

Original: D. D ATASETS USED IN S ECTION 3 IN THE MAIN TEXT
Revised: D. D ATASETS USED IN S ECTION 3 IN THE MAIN TEXT

Original: In Figure 1a in the main text, we use five challenging datasets, namely D1, D2, D3, D4, and D5, respectively.
Revised: In Figure 1a in the main text, we use five challenging datasets, namely D1, D2, D3, D4, and D5, respectively.

Original: Specifically, D1 represents MIPLIB mixed supportcase, D2 represents the single instance 𝛼 1 𝑇 𝑥 ≤ 𝛽𝑐𝑢𝑡 𝛼 2𝑇 𝑥 ≤ 𝛽𝛼 3𝑇 𝑥 ≤ 𝛽
Revised: Specifically, D1 represents MIPLIB mixed supportcase, D2 represents the single instance 𝛼 1 𝑇 𝑥 ≤ 𝛽𝑐𝑢𝑡 𝛼 2𝑇 𝑥 ≤ 𝛽𝛼 3𝑇 𝑥 ≤ 𝛽

Original: [Figure]
Revised: [Figure]

Original: 𝑐𝑢𝑡 𝑠𝑐𝑜𝑟𝑒1:0.8 𝑠𝑐𝑜𝑟𝑒2:0.78 𝑠𝑐𝑜𝑟𝑒3:0.
Revised: 𝑐𝑢𝑡 𝑠𝑐𝑜𝑟𝑒1:0.8 𝑠𝑐𝑜𝑟𝑒2:0.78 𝑠𝑐𝑜𝑟𝑒3:0.

Original: 𝛼 1 𝑇 𝑥 ≤ 𝛽𝑐𝑢𝑡 𝛼 2𝑇 𝑥 ≤ 𝛽𝑐𝑢𝑡 𝛼 1 𝑇 𝑥 ≤ 𝛽𝑐𝑢𝑡 𝛼 2 𝑇 𝑥 ≤ 𝛽𝛼 3 𝑇 𝑥 ≤ 𝛽𝑐𝑢𝑡 𝑐𝑢𝑡 𝛼 1𝑇 𝑥 ≤ 𝛽
Revised: 𝛼 1 𝑇 𝑥 ≤ 𝛽𝑐𝑢𝑡 𝛼 2𝑇 𝑥 ≤ 𝛽𝑐𝑢𝑡 𝛼 1 𝑇 𝑥 ≤ 𝛽𝑐𝑢𝑡 𝛼 2 𝑇 𝑥 ≤ 𝛽𝛼 3 𝑇 𝑥 ≤ 𝛽𝑐𝑢𝑡 𝑐𝑢𝑡 𝛼 1𝑇 𝑥 ≤ 𝛽

Original: [Figure]
Revised: [Figure]

Original: 3𝑇 𝑥 ≤ 𝛽
Revised: 3𝑇 𝑥 ≤ 𝛽

Original: [Figure]
Revised: [Figure]

Original: 𝑐𝑢𝑡 𝑒𝑛𝑑
Revised: 𝑐𝑢𝑡 𝑒𝑛𝑑

Original: Sequence model based policy
Revised: Sequence model based policy

Original: Score-based policy
Revised: Score-based policy

Original: [Figure]
Revised: [Figure]

Original: neos-1456979 from MIPLIB 2017, D3 represents MIPLIB mixed neos, D4 represents Anonymous, and D5 represents the single instance lectsched-5-obj from MIPLIB 2017.
Revised: neos-1456979 from MIPLIB 2017, D3 represents MIPLIB mixed neos, D4 represents Anonymous, and D5 represents the single instance lectsched-5-obj from MIPLIB 2017.

Original: In Figure 1b in the main text, we use the dataset MIPLIB mixed neos.
Revised: In Figure 1b in the main text, we use the dataset MIPLIB mixed neos.

Original: D. L ARGE - SCALE REAL - WORLD PRODUCTION PLANNING PROBLEMS
Revised: D. L ARGE - SCALE REAL - WORLD PRODUCTION PLANNING PROBLEMS

Original: The production planning problem aims to find the optimal production planning for thousands of factories according to the daily order demand.
Revised: The production planning problem aims to find the optimal production planning for thousands of factories according to the daily order demand.

Original: The constraints include the production capacity for each production line in each factory, transportation limit, the order rate, etc.
Revised: The constraints include the production capacity for each production line in each factory, transportation limit, the order rate, etc.

Original: The optimization objective is to minimize the production cost and lead time simultaneously.
Revised: The optimization objective is to minimize the production cost and lead time simultaneously.

Original: We split the dataset into training and test sets with 80% and 20% of the instances.
Revised: We split the dataset into training and test sets with 80% and 20% of the instances.

Original: The average size of the production planning problems is approximately equal to 3500 ×
Revised: The average size of the production planning problems is approximately equal to 3500 ×

Original: 5000 = 1 .
Revised: 5000 = 1 .

Original: 75 × 10 8 , which are large-scale real-world problems.
Revised: 75 × 10 8 , which are large-scale real-world problems.

Original: To promote the machine learning community for MILP, we will release the dataset once the paper is accepted to be published.
Revised: To promote the machine learning community for MILP, we will release the dataset once the paper is accepted to be published.

Original: E I LLUSTRATION OF A DVANTAGES OF U SING A S EQUENCE M ODEL
Revised: E I LLUSTRATION OF A DVANTAGES OF U SING A S EQUENCE M ODEL

Original: Figure 4 illustrate two major advantages of using the sequence model to learn cut selection.
Revised: Figure 4 illustrate two major advantages of using the sequence model to learn cut selection.

Original: First, the sequence model takes into account the order of selected cuts by modeling the selected cuts as an output sequence.
Revised: First, the sequence model takes into account the order of selected cuts by modeling the selected cuts as an output sequence.

Original: As shown in Figure 4, the order of cuts determined by the sequence model is better than the score-based method, thus leading to a better initial basis for solving the LP relaxation faster.
Revised: As shown in Figure 4, the order of cuts determined by the sequence model is better than the score-based method, thus leading to a better initial basis for solving the LP relaxation faster.

Original: Second, the sequence model captures the interaction among cuts, as it models the joint conditional probability of the selected cuts given an input sequence of the candidate cuts.
Revised: Second, the sequence model captures the interaction among cuts, as it models the joint conditional probability of the selected cuts given an input sequence of the candidate cuts.

Original: As shown in Figure 4, the sequence model selects cuts that complement each other nicely, thus leading to a more tightened
Revised: As shown in Figure 4, the sequence model selects cuts that complement each other nicely, thus leading to a more tightened

Original: LP relaxation and speeding up solving the MILP.
Revised: LP relaxation and speeding up solving the MILP.

Original: F A LGORITHM I MPLEMENTATION AND E XPERIMENTAL S ETTINGS F. D ESIGNED CUT FEATURES
Revised: F A LGORITHM I MPLEMENTATION AND E XPERIMENTAL S ETTINGS F. D ESIGNED CUT FEATURES

Original: (Suppose c denotes the objective coefficient.)
Revised: (Suppose c denotes the objective coefficient.)

Original: [Table]
Revised: [Table]

Original: Following
Revised: Following

Original: Huang et al.
Revised: Huang et al.

Original: ; Wesselmann & Stuhl (2012); Dey & Molinaro (2018b); Achterberg (2007), we design thirteen cut features for the cut selection task, such as the extent to which a cut is violated by the current LP solution and the proportion of non-zero coefficients of a cut.
Revised: ; Wesselmann & Stuhl (2012); Dey & Molinaro (2018b); Achterberg (2007), we design thirteen cut features for the cut selection task, such as the extent to which a cut is violated by the current LP solution and the proportion of non-zero coefficients of a cut.

Original: We present a detailed description of the designed cut features in Table 7.
Revised: We present a detailed description of the designed cut features in Table 7.

Original: We emphasize that we do not tune the cut features.
Revised: We emphasize that we do not tune the cut features.

Original: Therefore, it is promising to further improve our method by designing better cut features or using graph neural networks to learn better features in future work.
Revised: Therefore, it is promising to further improve our method by designing better cut features or using graph neural networks to learn better features in future work.

Original: F. I MPLEMENTATION DETAILS OF THE BASELINES
Revised: F. I MPLEMENTATION DETAILS OF THE BASELINES

Original: In this part, we present a detailed description of all the baselines used in this paper.
Revised: In this part, we present a detailed description of all the baselines used in this paper.

Original: We denote a cut by α T x ≤ β and the optimal solution of the current LP relaxation by x ∗ .
Revised: We denote a cut by α T x ≤ β and the optimal solution of the current LP relaxation by x ∗ .

Original: Throughout all experiments, we set the ratio of selected cuts as 0 .
Revised: Throughout all experiments, we set the ratio of selected cuts as 0 .

Original: 2 for all score-based rules and learning baselines.
Revised: 2 for all score-based rules and learning baselines.

Original: Random.
Revised: Random.

Original: Random selects a fixed ratio of the candidate cuts stochastically.
Revised: Random selects a fixed ratio of the candidate cuts stochastically.

Original: The ratio is set as 0 .
Revised: The ratio is set as 0 .

Original: 2 in this paper.
Revised: 2 in this paper.

Original: Normalized Violation (NV).
Revised: Normalized Violation (NV).

Original: NV is a score-based rule.
Revised: NV is a score-based rule.

Original: It scores each cut based on the normalized violation of the cut to the current LP solution, and selects a fixed ratio of cuts with high scores.
Revised: It scores each cut based on the normalized violation of the cut to the current LP solution, and selects a fixed ratio of cuts with high scores.

Original: The normalized violation is defined by max { 0 , α T x ∗ LP − β
Revised: The normalized violation is defined by max { 0 , α T x ∗ LP − β

Original: | β | } .
Revised: | β | } .

Original: The ratio is set as 0 .
Revised: The ratio is set as 0 .

Original: 2 in this paper.
Revised: 2 in this paper.

Original: Efficacy (Eff).
Revised: Efficacy (Eff).

Original: Eff is a score-based rule.
Revised: Eff is a score-based rule.

Original: It scores each cut based on the Euclidean distance of the cut hyperplane to the current LP solution, and selects a fixed ratio of cuts with high scores.
Revised: It scores each cut based on the Euclidean distance of the cut hyperplane to the current LP solution, and selects a fixed ratio of cuts with high scores.

Original: The ratio is set as 0 .
Revised: The ratio is set as 0 .

Original: 2 in this paper.
Revised: 2 in this paper.

Original: Default.
Revised: Default.

Original: Default is the default cut selection rule used in SCIP 8.0.0 (Bestuzheva et al., 2021).
Revised: Default is the default cut selection rule used in SCIP 8.0.0 (Bestuzheva et al., 2021).

Original: Please refer to (Achterberg, 2007) for a detailed description of the SCIP’s default cut selection rule.
Revised: Please refer to (Achterberg, 2007) for a detailed description of the SCIP’s default cut selection rule.

Original: Note that Default tackles the two problems: (1) which cuts should be preferred, and (2) how many cuts should be selected, in cut selection by human-designed heuristics.
Revised: Note that Default tackles the two problems: (1) which cuts should be preferred, and (2) how many cuts should be selected, in cut selection by human-designed heuristics.

Original: That is, Default selects variable ratios of cuts rather than a fixed ratio.
Revised: That is, Default selects variable ratios of cuts rather than a fixed ratio.

Original: Score-based policy (SBP).
Revised: Score-based policy (SBP).

Original: Since the state-of-the-art (SOTA) reinforcement learning based method for cut selection (Tang et al., 2020) is designed for the setting that selects the best cut in each round, we implement a slight variant of the SOTA to adapt to our setting that selects a subset of cuts in each round, namely SBP.
Revised: Since the state-of-the-art (SOTA) reinforcement learning based method for cut selection (Tang et al., 2020) is designed for the setting that selects the best cut in each round, we implement a slight variant of the SOTA to adapt to our setting that selects a subset of cuts in each round, namely SBP.

Original: Specifically, the core idea of SBP is learning a scoring function to measure cut quality as Tang et al.
Revised: Specifically, the core idea of SBP is learning a scoring function to measure cut quality as Tang et al.

Original: ; Huang et al.
Revised: ; Huang et al.

Original: ; Paulus et al.
Revised: ; Paulus et al.

Original: (2022) do.
Revised: (2022) do.

Original: For a fair comparison, SBP uses the same cut features as HEM and we train SBP via reinforcement learning as well.
Revised: For a fair comparison, SBP uses the same cut features as HEM and we train SBP via reinforcement learning as well.

Original: Our implemented SBP is also a slight variant of the method proposed in Huang et al.
Revised: Our implemented SBP is also a slight variant of the method proposed in Huang et al.

Original: We emphasize that experiments in the main text show that our implemented SBP is a strong baseline.
Revised: We emphasize that experiments in the main text show that our implemented SBP is a strong baseline.

Original: Specifically, we implement the scoring function with a multi-layer perceptron that predicts the score of a given cut.
Revised: Specifically, we implement the scoring function with a multi-layer perceptron that predicts the score of a given cut.

Original: That is, the scoring function predicts a cut’s score based on the features of the cut.
Revised: That is, the scoring function predicts a cut’s score based on the features of the cut.

Original: The MLP network contains two hidden layers with 128 units.
Revised: The MLP network contains two hidden layers with 128 units.

Original: Moreover, we train the scoring function via evolutionary strategies as (Tang et al., 2020) does.
Revised: Moreover, we train the scoring function via evolutionary strategies as (Tang et al., 2020) does.

Original: We will also release the code for implementing SBP once the paper is accepted to be published.
Revised: We will also release the code for implementing SBP once the paper is accepted to be published.

Original: do do 5:
Revised: do do 5:

Original: D 6:
Revised: D 6:

Original: Take action a state s policy π 7:
Revised: Take action a state s policy π 7:

Original: Receive reward add a to D train 8: end for 9:
Revised: Receive reward add a to D train 8: end for 9:

Original: Compute policy
Revised: Compute policy

Original: D train
Revised: D train

Original: Update the parameters, θ α
Revised: Update the parameters, θ α

Original: 11: end for
Revised: 11: end for

Original: F
Revised: F

Original:  I MPLEMENTATION DETAILS AND HYPERPARAMETERS F.3. H ARDWARE SPECIFICATION
Revised:  I MPLEMENTATION DETAILS AND HYPERPARAMETERS F.3. H ARDWARE SPECIFICATION

Original: Throughout all experiments, we use a single machine that contains eight GPU devices (NVidia
Revised: Throughout all experiments, we use a single machine that contains eight GPU devices (NVidia

Original: GeForce GTX 3090 Ti) and two Intel Gold 6246R CPUs.
Revised: GeForce GTX 3090 Ti) and two Intel Gold 6246R CPUs.

Original: F.3. S OLVER SETUP
Revised: F.3. S OLVER SETUP

Original: For reproducibility, we emphasize that all results in the main text are obtained by averaging results over the SCIP random seeds { 1 , 2 , 3 } .
Revised: For reproducibility, we emphasize that all results in the main text are obtained by averaging results over the SCIP random seeds { 1 , 2 , 3 } .

Original: F.3. R EWARD FUNCTION
Revised: F.3. R EWARD FUNCTION

Original: [Table]
Revised: [Table]

Original: On the easy datasets, we set the reward as the negative solving time.
Revised: On the easy datasets, we set the reward as the negative solving time.

Original: On the medium and hard datasets, we set the reward as the negative primaldual gap integral within a time limit of 300 seconds.
Revised: On the medium and hard datasets, we set the reward as the negative primaldual gap integral within a time limit of 300 seconds.

Original: For the real-world production planning problems, we set the reward as the negative primal-dual gap integral within a time limit of 600 seconds or the negative dual bound improvement.
Revised: For the real-world production planning problems, we set the reward as the negative primal-dual gap integral within a time limit of 600 seconds or the negative dual bound improvement.

Original: The results reported in the main text are achieved by HEM with the negative dual bound improvement reward.
Revised: The results reported in the main text are achieved by HEM with the negative dual bound improvement reward.

Original: We provide the performance of HEM that uses the negative primal-dual gap integral in Table 8.
Revised: We provide the performance of HEM that uses the negative primal-dual gap integral in Table 8.

Original: The results still show that HEM significantly outperforms all the baselines in terms of the Time and PD integral.
Revised: The results still show that HEM significantly outperforms all the baselines in terms of the Time and PD integral.

Original: We emphasize that we can set the reward according to our objective in real-world problems.
Revised: We emphasize that we can set the reward according to our objective in real-world problems.

Original: For example, suppose we aim to minimize the primal-dual gap within a time limit, then we can set the reward as the primal-dual gap within the time limit.
Revised: For example, suppose we aim to minimize the primal-dual gap within a time limit, then we can set the reward as the primal-dual gap within the time limit.

Original: F.3. P OLICY NETWORK ARCHITECTURE
Revised: F.3. P OLICY NETWORK ARCHITECTURE

Original: The higher-level model contains an LSTM encoder and an MLP.
Revised: The higher-level model contains an LSTM encoder and an MLP.

Original: The LSTM network encodes variable-sized inputs into hidden vectors with dimension 128.
Revised: The LSTM network encodes variable-sized inputs into hidden vectors with dimension 128.

Original: The MLP network contains two hidden layers with 128 units.
Revised: The MLP network contains two hidden layers with 128 units.

Original: The lower-level model is essentially a pointer network.
Revised: The lower-level model is essentially a pointer network.

Original: We keep the hyperparameters of the pointer network as that used in (Bello* et al., 2017). 
Revised: We keep the hyperparameters of the pointer network as that used in (Bello* et al., 2017). [Figure]

Original: F.3. O PTIMIZATION
Revised: F.3. O PTIMIZATION

Original: Throughout all experiments, we apply Adam optimizer with learning rate α 1 = 1 × 10 − 4 to optimize the lower-level model, and learning rate α 2 = 5 × 10 − 4 to optimize the higher-level model.
Revised: Throughout all experiments, we apply Adam optimizer with learning rate α 1 = 1 × 10 − 4 to optimize the lower-level model, and learning rate α 2 = 5 × 10 − 4 to optimize the higher-level model.

Original: For each epoch, we collect 32 samples for training, and we set the total epochs as 100.
Revised: For each epoch, we collect 32 samples for training, and we set the total epochs as 100.

Original: It is surprising that learning a good cut selection policy does not need too much data as shown in Tang et al.
Revised: It is surprising that learning a good cut selection policy does not need too much data as shown in Tang et al.

Original: For training stability, we delay the higher-level policy update.
Revised: For training stability, we delay the higher-level policy update.

Original: This creates a two-timescale algorithm, as often required for convergence in the linear setting (Fujimoto et al.,  Konda & Tsitsiklis, 2003).
Revised: This creates a two-timescale algorithm, as often required for convergence in the linear setting (Fujimoto et al., 2018; Konda & Tsitsiklis, 2003).

Original: We set the delay update freq as two.
Revised: We set the delay update freq as two.

Original: That is, we first train the lower-level policy twice, then train the higher-level policy once.
Revised: That is, we first train the lower-level policy twice, then train the higher-level policy once.

Original: Additionally, the results in Appendix G.3.4 show that the convergence performance of HEM is insensitive to the hyperparameter delay update freq.
Revised: Additionally, the results in Appendix G.3.4 show that the convergence performance of HEM is insensitive to the hyperparameter delay update freq.

Original: F.3. T HE TRAINING ALGORITHM
Revised: F.3. T HE TRAINING ALGORITHM

Original: We provide the procedure of the training algorithm of HEM in Algorithm 2.
Revised: We provide the procedure of the training algorithm of HEM in Algorithm 2.

Original: F. M ORE DETAILS OF HEM F.4. D ETAILS OF THE POINTER NETWORK
Revised: F. M ORE DETAILS OF HEM F.4. D ETAILS OF THE POINTER NETWORK

Original: The pointer network is first introduced by (Vinyals et al., 2015) for directly finding solutions of specific combinatorial optimization problems, such as the Travelling Salesman Problems.
Revised: The pointer network is first introduced by (Vinyals et al., 2015) for directly finding solutions of specific combinatorial optimization problems, such as the Travelling Salesman Problems.

Original: The pointer network architecture is illustrated in Figure 5.
Revised: The pointer network architecture is illustrated in Figure 5.

Original: The pointer network consists of a Long Short-Term
Revised: The pointer network consists of a Long Short-Term

Original: Memory encoder, a Long Short-Term Memory decoder, and an attention that is used as a pointer to select a member of the input sequence as the output (Vinyals et al., 2015).
Revised: Memory encoder, a Long Short-Term Memory decoder, and an attention that is used as a pointer to select a member of the input sequence as the output (Vinyals et al., 2015).

Original: Specifically, we implement the pointer network following Bello* et al.
Revised: Specifically, we implement the pointer network following Bello* et al.

Original: Please refer to (Bello* et al., 2017) for implementation details of the pointer network.
Revised: Please refer to (Bello* et al., 2017) for implementation details of the pointer network.

Original: The major difference between our used pointer network and the pointer network used in (Bello* et al., 2017) is that we use the pointer network to select ordered subsets of input sequences, but (Bello* et al., 2017) use the pointer network to output permutations of input sequences.
Revised: The major difference between our used pointer network and the pointer network used in (Bello* et al., 2017) is that we use the pointer network to select ordered subsets of input sequences, but (Bello* et al., 2017) use the pointer network to output permutations of input sequences.

Original: F.4. D IFFERENCE BETWEEN HEM- RATIO AND HEM W / O H
Revised: F.4. D IFFERENCE BETWEEN HEM- RATIO AND HEM W / O H

Original: Details of HEM w/o H To implement HEM w/o H, we augment each input sequence with an end token, i.e., a thirteen-dimensional tensor with values all being one.
Revised: Details of HEM w/o H To implement HEM w/o H, we augment each input sequence with an end token, i.e., a thirteen-dimensional tensor with values all being one.

Original: The end token is at the end position of the input sequence.
Revised: The end token is at the end position of the input sequence.

Original: Once the decoder of HEM w/o H outputs the end token, then the decoding ends.
Revised: Once the decoder of HEM w/o H outputs the end token, then the decoding ends.

Original: That is, HEM w/o H can implicitly predict the number of cuts that should be selected by predicting whether to decode the end token at the current decoding step.
Revised: That is, HEM w/o H can implicitly predict the number of cuts that should be selected by predicting whether to decode the end token at the current decoding step.

Original: The policy network of HEM-ratio and HEM w/o H are both essentially a pointer network (Vinyals et al., 2015), a variant of the sequence model.
Revised: The policy network of HEM-ratio and HEM w/o H are both essentially a pointer network (Vinyals et al., 2015), a variant of the sequence model.

Original: We present the major difference between HEM-ratio[Table] ) and HEM w/o H in the following.
Revised: We present the major difference between HEM-ratio and HEM w/o H in the following.

Original: HEM w/o H predicts an end token as used in language tasks (Sutskever et al., 2014; Vaswani et al., 2017) to determine the number of cuts that should be selected implicitly.
Revised: HEM w/o H predicts an end token as used in language tasks (Sutskever et al., 2014; Vaswani et al., 2017) to determine the number of cuts that should be selected implicitly.

Original: In contrast, HEM-ratio always selects a fixed ratio of cuts, i.e, it always ends decoding  at a pre-determined position.
Revised: In contrast, HEM-ratio always selects a fixed ratio of cuts, i.e, it always ends decoding [Table] ) at a pre-determined position.

Original: Therefore, HEM w/o H can learn the number of cuts that should be selected, but HEM-ratio cannot.
Revised: Therefore, HEM w/o H can learn the number of cuts that should be selected, but HEM-ratio cannot.

Original: F.4. M ORE DISCUSSION OF HEM
Revised: F.4. M ORE DISCUSSION OF HEM

Original: In this part, we provide details of some more advantages of HEM.
Revised: In this part, we provide details of some more advantages of HEM.

Original: (1) Inspired by hierarchical reinforcement learning (Sutton et al., 1999b; Nachum et al., 2018), HEM leverages the hierarchical structure of the cut selection task, which is important for efficient exploration in complex decisionmaking tasks.
Revised: (1) Inspired by hierarchical reinforcement learning (Sutton et al., 1999b; Nachum et al., 2018), HEM leverages the hierarchical structure of the cut selection task, which is important for efficient exploration in complex decisionmaking tasks.

Original: (2) Previous methods (Tang et al., 2020; Huang et al., 2022) usually train cut selection policies via black-box optimization methods such as evolution strategies (Salimans et al., 2017).
Revised: (2) Previous methods (Tang et al., 2020; Huang et al., 2022) usually train cut selection policies via black-box optimization methods such as evolution strategies (Salimans et al., 2017).

Original: In contrast, HEM is differentiable and we train the HEM via gradient-based algorithms, which is more sample efficient than black-box optimization methods (Sutton & Barto, 2018; Schulman et al., 2015).
Revised: In contrast, HEM is differentiable and we train the HEM via gradient-based algorithms, which is more sample efficient than black-box optimization methods (Sutton & Barto, 2018; Schulman et al., 2015).

Original: Although we can offline generate training samples as much as possible using a MILP solver, high sample efficiency is significant as generating samples can be extremely time-consuming in practice.
Revised: Although we can offline generate training samples as much as possible using a MILP solver, high sample efficiency is significant as generating samples can be extremely time-consuming in practice.

Original: G M ORE R ESULTS G. M ORE MOTIVATING RESULTS
Revised: G M ORE R ESULTS G. M ORE MOTIVATING RESULTS

Original: Ratio matters.
Revised: Ratio matters.

Original: To evaluate the effect of the ratio of selected cuts on solving MILPs, we focus on the
Revised: To evaluate the effect of the ratio of selected cuts on solving MILPs, we focus on the

Original: Normalized Violation (NV) cut selection method with different ratios of selected cuts.
Revised: Normalized Violation (NV) cut selection method with different ratios of selected cuts.

Original: (1) We first evaluate the NV methods that select 5% , 10% , 20% , 30% , 40% , 50% , 60% , 70% , and 80% of candidate cuts, respectively, on four datasets.
Revised: (1) We first evaluate the NV methods that select 5% , 10% , 20% , 30% , 40% , 50% , 60% , 70% , and 80% of candidate cuts, respectively, on four datasets.

Original: The results in Figure 6a show that the NV achieves better solver performance with larger ratios on CORLAT and MIPLIB mixed neos.
Revised: The results in Figure 6a show that the NV achieves better solver performance with larger ratios on CORLAT and MIPLIB mixed neos.

Original: The results demonstrate that the ratio that leads to better solver performance is variable over different datasets, which implies that learning dataset-dependent ratios is important.
Revised: The results demonstrate that the ratio that leads to better solver performance is variable over different datasets, which implies that learning dataset-dependent ratios is important.

Original: (2) We then evaluate the NV methods that select 5% , 10% , 20% , 30% , 40% , 50% , 60% , 70% , and 80% of candidate cuts, respectively, on four instances from the Anonymous dataset.
Revised: (2) We then evaluate the NV methods that select 5% , 10% , 20% , 30% , 40% , 50% , 60% , 70% , and 80% of candidate cuts, respectively, on four instances from the Anonymous dataset.

Original: The results in Figure 6b show that NV achieves better 
Revised: The results in Figure 6b show that NV achieves better solver performance with larger ratios on Anonymous 121 and Anonymous 131.

Revised: The results demonstrate that the ratio that leads to better solver performance is variable over different instances from the same dataset, which implies that learning instance-dependent ratios is important as well.

Original: [Figure]
Revised: [Figure]

Original: li z
Revised: r

Original: [Figure]
Revised: [Figure]

Original: p r i m a l ­
Revised: a li z

Original: [Figure]
Revised: [Figure]

Original:  i n 
Revised: p r i m a l ­ dua l gap i n t

Original: [Figure]
Revised: [Figure]

Original: solver performance with larger ratios on Anonymous 121 and Anonymous 131.

Original: The results demonstrate that the ratio that leads to better solver performance is variable over different instances from the same dataset, which implies that learning instance-dependent ratios is important as well.

Original: G. M ORE RESULTS OF MAIN EVALUATION
Revised: G. M ORE RESULTS OF MAIN EVALUATION

Original: In this section, we provide more results of the main evaluation.
Revised: In this section, we provide more results of the main evaluation.

Original: The results in Table 9 show the performance of HEM and the baselines in terms of the total number of nodes (Nodes) and the primal-dual gap (PD gap).
Revised: The results in Table 9 show the performance of HEM and the baselines in terms of the total number of nodes (Nodes) and the primal-dual gap (PD gap).

Original: (1) Easy datasets.
Revised: (1) Easy datasets.

Original: On the easy datasets, HEM and most baselines find the optimal solution within the time limit, as the PD gap converges to zero.
Revised: On the easy datasets, HEM and most baselines find the optimal solution within the time limit, as the PD gap converges to zero.

Original: Additionally,
Revised: Additionally,

Original: HEM significantly outperforms all the baselines in terms of the Nodes on the easy datasets.
Revised: HEM significantly outperforms all the baselines in terms of the Nodes on the easy datasets.

Original: Medium and hard datasets.
Revised: Medium and hard datasets.

Original: In terms of the PD gap, HEM outperforms all the baselines on medium and hard datasets, especially on CORLAT.
Revised: In terms of the PD gap, HEM outperforms all the baselines on medium and hard datasets, especially on CORLAT.

Original: However, the Nodes metric cannot well distinguish the performance of different cut selection methods on the medium and hard datasets for the following two reasons.
Revised: However, the Nodes metric cannot well distinguish the performance of different cut selection methods on the medium and hard datasets for the following two reasons.

Original: First, the solving time of the LP relaxation on each node is different, and thus the
Revised: First, the solving time of the LP relaxation on each node is different, and thus the

Original: Nodes cannot directly determine the solving time (Huang et al., 2022).
Revised: Nodes cannot directly determine the solving time (Huang et al., 2022).

Original: Second, on those unsolved instances within the time limit, the Nodes metric is not a proper metric, as the Nodes cannot evaluate the quality of the solving process.
Revised: Second, on those unsolved instances within the time limit, the Nodes metric is not a proper metric, as the Nodes cannot evaluate the quality of the solving process.

Original: G. M ORE RESULTS OF ABLATION STUDY
Revised: G. M ORE RESULTS OF ABLATION STUDY

Original: In this section, we provide more results of ablation studies in the main text
Revised: In this section, we provide more results of ablation studies in the main text.

Original: . G.3. I N - DEPTH ANALYSIS OF HEM- RATIO AND SBP
Revised: G.3. I N - DEPTH ANALYSIS OF HEM- RATIO AND SBP

Original: We provide possible reasons for HEM-ratio performing poorer than SBP on several challenging
Revised: We provide possible reasons for HEM-ratio performing poorer than SBP on several challenging

Original: MILP problem benchmarks.
Revised: MILP problem benchmarks.

Original: Fundamentally, HEM-ratio formulates the cut selection task as a sequence modeling problem, which has two main advantages over SBP.
Revised: Fundamentally, HEM-ratio formulates the cut selection task as a sequence modeling problem, which has two main advantages over SBP.

Original: That is, the sequence model can not only capture the underlying order information, but also capture the interaction among cuts.
Revised: That is, the sequence model can not only capture the underlying order information, but also capture the interaction among cuts.

Original: However, training a sequence model is more difficult than training a scoring function, as the sequence model aims to learn a much more complex task.
Revised: However, training a sequence model is more difficult than training a scoring function, as the sequence model aims to learn a much more complex task.

Original: Specifically, the scoring function aims to learn to score each cut, while the sequence model aims to model the joint probability of the selected cuts.
Revised: Specifically, the scoring function aims to learn to score each cut, while the sequence model aims to model the joint probability of the selected cuts.

Original: The latter is a more challenging learning task.
Revised: The latter is a more challenging learning task.

Original: Moreover, we follow the reinforcement learning paradigm instead of supervised learning to train the model, making the training process more unstable.
Revised: Moreover, we follow the reinforcement learning paradigm instead of supervised learning to train the model, making the training process more unstable.

Original: Therefore, the sequence model may suffer from inefficient exploration and be trapped to a local optimum.
Revised: Therefore, the sequence model may suffer from inefficient exploration and be trapped to a local optimum.

Original: As a result, HEM-ratio can perform poorer than SBP, especially on challenging
Revised: As a result, HEM-ratio can perform poorer than SBP, especially on challenging

Original: MILP problem benchmarks.
Revised: MILP problem benchmarks.

Original: G.3. C ONTRIBUTION OF EACH COMPONENT
Revised: G.3. C ONTRIBUTION OF EACH COMPONENT

Original: To understand the contribution of each component of HEM, we provide more results of HEM and
Revised: To understand the contribution of each component of HEM, we provide more results of HEM and

Original: HEM without the higher-level model on Set Covering, Multiple Knapsack, MIK, Load Balancing,
Revised: HEM without the higher-level model on Set Covering, Multiple Knapsack, MIK, Load Balancing,

Original: Anonymous, and MIPLIB mixed supportcase.
Revised: Anonymous, and MIPLIB mixed supportcase.

Original: The results in Table 10 show that HEM outperforms
Revised: The results in Table 10 show that HEM outperforms

Revised: HEM w/o H in terms of the solving time, the primal-dual gap, and the primal-dual gap integral on several challenging datasets, demonstrating the importance of our proposed higher- model.

Revised: Moreover, HEM w/o H significantly outperforms SBP in terms of the solving time, the primal-dual

Original: [Table] [Table]
Revised: [Table] [Table]

Original: HEM w/o H in terms of the solving time, the primal-dual gap, and the primal-dual gap integral on several challenging datasets, demonstrating the importance of our proposed higher-level model.
Revised: gap, and the primal-dual gap integral on several challenging datasets, demonstrating the significance of our proposed lower-level model

Original: Moreover, HEM w/o H significantly outperforms SBP in terms of the solving time, the primal-dual gap, and the primal-dual gap integral on several challenging datasets, demonstrating the significance of our proposed lower-level model.
Revised: [Table]

Original: G.3. T HE IMPORTANCE OF TACKLING P1-P3 IN CUT SELECTION
Revised: G.3. T HE IMPORTANCE OF TACKLING P1-P3 IN CUT SELECTION

Original: To understand the importance of tackling P1-P3 in cut selection, we provide more results of HEM,
Revised: To understand the importance of tackling P1-P3 in cut selection, we provide more results of HEM,

Original: HEM-ratio, and HEM-ratio-order on Set Covering, Multiple Knapsack, MIK, Load Balancing,
Revised: HEM-ratio, and HEM-ratio-order on Set Covering, Multiple Knapsack, MIK, Load Balancing,

Original: Anonymous, and MIPLIB mixed supportcase.
Revised: Anonymous [Table] and MIPLIB mixed supportcase.

Original: Here we refresh what HEM, HEM-ratio, HEM-ratioorder mean.
Revised: Here we refresh what HEM, HEM-ratio, HEM-ratioorder mean.

Original: (1) HEM .
Revised: (1) HEM .

Original: HEM tackles P1-P3 in cut selection simultaneously.
Revised: HEM tackles P1-P3 in cut selection simultaneously.

Original: (2) HEM-ratio .
Revised: (2) HEM-ratio .

Original: In order not to learn how many cuts should be selected, we remove the higher-level model of HEM and force the lower-level model to select a fixed ratio of cuts.
Revised: In order not to learn how many cuts should be selected, we remove the higher-level model of HEM and force the lower-level model to select a fixed ratio of cuts.

Original: We denote it by HEM-ratio.
Revised: We denote it by HEM-ratio.

Original: Note that HEM-ratio is different from HEM w/o H (see Appendix F).
Revised: Note that HEM-ratio is different from HEM w/o H (see Appendix F).

Original: HEM-ratio tackles P1 and P3 in cut selection.
Revised: HEM-ratio tackles P1 and P3 in cut selection.

Original: (3) HEM-ratio-order.
Revised: (3) HEM-ratio-order.

Original: To further mute the effect of the order of selected cuts, we reorder the selected cuts given by HEM-ratio with the original index of the generated cuts, which we denote by HEM-ratio-order.
Revised: To further mute the effect of the order of selected cuts, we reorder the selected cuts given by HEM-ratio with the original index of the generated cuts, which we denote by HEM-ratio-order.

Original: HEM-ratio-order mainly tackles P1 .
Revised: HEM-ratio-order mainly tackles P1 .

Original: The results in Table 11 suggest the following.
Revised: The results in Table 11 suggest the following.

Original: HEM-ratio-order outperforms Default and NoCuts on several datasets, demonstrating that tackling P1 by data-driven methods is crucial.
Revised: HEM-ratio-order outperforms Default and NoCuts on several datasets, demonstrating that tackling P1 by data-driven methods is crucial.

Original: HEM signif-
Revised: HEM significantly outperforms HEM-ratio in terms of the primal-dual gap integral, demonstrating the significance of tackling P2 . HEM-ratio outperforms HEM-ratio-order on several datasets, which demonstrates the importance of tackling P3 . Moreover, HEM-ratio performs better than SBP in terms of

Original: [Table] [Table]
Revised: [Table] [Table]

Original: icantly outperforms HEM-ratio in terms of the primal-dual gap integral, demonstrating the significance of tackling P2 .
Revised: the solving time and the primal-dual gap integral on all six datasets except Set Covering and MIPLIB mixed supportcase

Original: HEM-ratio outperforms HEM-ratio-order on several datasets, which demonstrates the importance of tackling P3 .
Revised: [Table]

Original: Moreover, HEM-ratio performs better than SBP in terms of the solving time and the primal-dual gap integral on all six datasets except Set Covering and MIPLIB mixed supportcase, which shows the superiority of formulating the cut selection as a sequence to sequence learning problem over formulating it as a scoring task.
Revised: which shows the superiority of formulating the cut selection as a sequence to sequence learning problem over formulating it as a scoring task.

Original: However, HEM-ratio and HEMratio-order perform a little worse than SBP on Set Covering and MIPLIB mixed supportcase.
Revised: However, HEM-ratio and HEMratio-order perform a little worse than SBP on Set Covering and MIPLIB mixed supportcase.

Original: A possible reason is that it is more difficult to train a sequence model than to train a multi-layer perceptron and thus the sequence model may suffer from inefficient exploration and be trapped to the local optimum.
Revised: A possible reason is that it is more difficult to train a sequence model than to train a multi-layer perceptron and thus the sequence model may suffer from inefficient exploration and be trapped to the local optimum.

Original: Please refer to Appendix G.3.1 for a detailed analysis.
Revised: Please refer to Appendix G.3.1 for a detailed analysis.

Original: G.
Revised: G.3. S ENSITIVITY ANALYSIS

Original: . S ENSITIVITY ANALYSIS

Original: Additionally, we perform ablation studies to test the sensitivity of HEM to the hyperparameter delay update freq d .
Revised: Additionally, we perform ablation studies to test the sensitivity of HEM to the hyperparameter delay update freq d .

Original: The results in Table 12 show that there is a wide range of d for HEM to achieve comparable performance on Maximum Independent Set, CORLAT, and MIPLIB mixed neos.
Revised: The results in Table 12 show that there is a wide range of d for HEM to achieve comparable performance on Maximum Independent Set, CORLAT, and MIPLIB mixed neos.

Original: Moreover, we emphasize that we do not tune the hyperparameter d .
Revised: Moreover, we emphasize that we do not tune the hyperparameter d .

Original: As the results shown in Table 12, d = 3 and d = 4 performs the best on CORLAT and MIPLIB mixed neos, respectively.
Revised: As the results shown in Table 12, d = 3 and d = 4 performs the best on CORLAT and MIPLIB mixed neos, respectively.

Original: However, we simply set d = 2 throughout all experiments in the main text.
Revised: However, we simply set d = 2 throughout all experiments in the main text.

Original: G. M ORE RESULTS OF GENERALIZATION
Revised: G. M ORE RESULTS OF GENERALIZATION

Original: Here we provide more results of the generalization experiments on Set Covering.
Revised: Here we provide more results of the generalization experiments on Set Covering.

Original: On Set Covering, we test HEM on two times and four times larger instances than those seen during training.
Revised: On Set Covering, we test HEM on two times and four times larger instances than those seen during training.

Original: The results in Table 13 show that HEM generalizes well to instances that are significantly larger than seen during training.
Revised: The results in Table 13 show that HEM generalizes well to instances that are significantly larger than seen during training.

Original: In particular, HEM achieves at least 70% improvement in terms of the Time compared to all the rule-based baselines.
Revised: In particular, HEM achieves at least 70% improvement in terms of the Time compared to all the rule-based baselines.

Original: Moreover, SBP also generalizes well to large instances, demonstrating that SBP is a strong baseline.
Revised: Moreover, SBP also generalizes well to large instances, demonstrating that SBP is a strong baseline.

Original: G. M ORE VISUALIZATION RESULTS
Revised: G. M ORE VISUALIZATION RESULTS

Original: In this part, we provide more visualization results on Set Covering and MIK.
Revised: In this part, we provide more visualization results on Set Covering and MIK.

Original: On MIK, we visualize the cuts selected by HEM-ratio and SBP on a randomly sampled instance.
Revised: On MIK, we visualize the cuts selected by HEM-ratio and SBP on a randomly sampled instance.

Original: We perform principal component analysis (Mohri et al., 2018) on selected cuts to reduce the cut features to two-dimensional space.
Revised: We perform principal component analysis (Mohri et al., 2018) on selected cuts to reduce the cut features to two-dimensional space.

Original: Colored points illustrate reduced cut features.
Revised: Colored points illustrate reduced cut features.

Original: To visualize the diversity of selected cuts, we use dashed lines to connect the points with the smallest and largest x,y coordinates.
Revised: To visualize the diversity of selected cuts, we use dashed lines to connect the points with the smallest and largest x,y coordinates.

Original: The results in Figure 7 still show that HEM-ratio selects much more diverse cuts than SBP on MIK.
Revised: The results in Figure 7 still show that HEM-ratio selects much more diverse cuts than SBP on MIK.

Original: However, HEM-ratio performs poorer than SBP on Set Covering (see Appendix G.3.1 for a detailed analysis).
Revised: However, HEM-ratio performs poorer than SBP on Set Covering (see Appendix G.3.1 for a detailed analysis).

Original: Therefore, we visualize the cuts selected by HEM and SBP on a randomly sampled instance from Set Covering.
Revised: Therefore, we visualize the cuts selected by HEM and SBP on a randomly sampled instance from Set Covering.

Original: Although HEM learns the number of cuts that should be selected, we find that HEM selects much fewer cuts than SBP.
Revised: Although HEM learns the number of cuts that should be selected, we find that HEM selects much fewer cuts than SBP.

Original: Specifically, HEM selects 25 cuts, while SBP selects 158 cuts.
Revised: Specifically, HEM selects 25 cuts, while SBP selects 158 cuts.

Original: Interestingly, the results in Figure 7 show that SBP selects 158 similar cuts with high 
Revised: Interestingly, the results in Figure 7 show that SBP selects 158 similar cuts with high scores, while HEM selects much more diverse cuts than SBP. The results show that HEM tends to select cuts that complement each other nicely.

Original: [Figure]
Revised: [Figure]

Original: Set Covering
Revised: Set Covering

Original: SBP
Revised: SBP

Original: HEM
Revised: HEM

Original: [Figure]
Revised: [Figure]

Original: scores, while HEM selects much more diverse cuts than SBP. The results show that HEM tends to select cuts that complement each other nicely.
Revised: [Table]

Original: G. E VALUATION WITH A TIME LIMIT OF THREE HOURS
Revised: G. E VALUATION WITH A TIME LIMIT OF THREE HOURS

Original: In this section, we aim to evaluate whether HEM can generalize well to solving problems within a much longer time limit.
Revised: In this section [Table] we aim to evaluate whether HEM can generalize well to solving problems within a much longer time limit.

Original: Specifically, we evaluate HEM on two extremely challenging MIPLIB datasets within a time limit of three hours .
Revised: Specifically, we evaluate HEM on two extremely challenging MIPLIB datasets within a time limit of three hours .

Original: Note that we still train HEM with a time limit ofseconds, while we test HEM with a time limit of three hours.
Revised: Note that we still train HEM with a time limit ofseconds, while we test HEM with a time limit of three hours.

Original: The results in Table 14 show that
Revised: The results in Table 14 show that

Original: HEM still significantly outperforms all the baselines, especially in terms of the primal-dual gap integral on MIPLIB mixed neos and MIPLIB mixed supportcase.
Revised: HEM still significantly outperforms all the baselines, especially in terms of the primal-dual gap integral on MIPLIB mixed neos and MIPLIB mixed supportcase.

Original: In terms of the primal-dual gap,
Revised: In terms of the primal-dual gap,

Original: HEM also outperforms the baselines.
Revised: HEM also outperforms the baselines.

Original: Moreover, HEM performs better than baselines in terms of the solving time on MIPLIB mixed neos, but HEM performs poorly in terms of the solving time on
Revised: Moreover, HEM performs better than baselines in terms of the solving time on MIPLIB mixed neos, but HEM performs poorly in terms of the solving time on

Original: MIPLIB mixed supportcase.
Revised: MIPLIB mixed supportcase.

Original: Interestingly, the primal-dual gap integral is not always consistent with the solving time.
Revised: Interestingly, the primal-dual gap integral is not always consistent with the solving time.

Original: We emphasize that we train with the negative primal-dual gap integral reward.
Revised: We emphasize that we train with the negative primal-dual gap integral reward.

Original: To further improve the performance of HEM in terms of the solving time, we can set the reward as the negative solving time instead of the negative primal-dual gap integral.
Revised: To further improve the performance of HEM in terms of the solving time, we can set the reward as the negative solving time instead of the negative primal-dual gap integral.

Original: G. T RAINING CURVES
Revised: G. T RAINING CURVES

Original: In this section, we provide the training curves of HEM on all nine datasets.
Revised: In this section, we provide the training curves of HEM on all nine datasets.

Original: The results in Figure 8 show that the performance of our learned policies in terms of the solving time or the primal-dual gap integral drops with the training epochs, demonstrating the effectiveness of our learning process.
Revised: The results in Figure 8 show that the performance of our learned policies in terms of the solving time or the primal-dual gap integral drops with the training epochs, demonstrating the effectiveness of our learning process.

Original: [Figure]
Revised: Our proposed HEM can well generalize to the multiple rounds setting. We describe the formulation for the multiple rounds setting in detail, and conduct the experiments under this setting as follows. The results show that our proposed HEM outperforms the baselines in terms of the Time (up to 97.63% improvement) and Primal-dual gap integral (up to 34.07% improvement).

Original: t r a i n i n g / p r i m a l - d u a l g a p
Revised: Formulation for the multiple rounds setting. As mentioned in Section 4, we formulate the cut selection problem as a Markov decision process, which is applicable to the multiple rounds setting. Specifically, under the multiple rounds setting, we keep the state space, action space, reward, and transition function unchanged, and set the cut separation rounds as T > 1 .

Original: i n t e g r a l
Revised: We have described the formulation for the multiple rounds setting in detail in the revised version. Experiments under the multiple rounds setting. HEM can well generalize to the multiple rounds setting as shown in Table 15. The table shows that HEM—that is trained in the multiple rounds

Original: [Figure]
Revised: [Figure]

Original: 1e
Revised: t r a i n i n g / p r i m a l - d u a l g a p

Original: [Figure]
Revised: i n t e g r a l

Original: [Figure]
Revised: Anonymous HEM 0.

Original:      t r a i n i n g / p r i m a l - d u a l g a p
Revised: 1.0 Epoch 1e2     t r a i n i n g / p r i m a l - d u a l g a p

Original: i n t e g r a l
Revised: i n t e g r a l

Revised: MIPLIB mixed neos HEM 0.

Revised: 1.0 Epoch 1e       t r a i n i n g / p r i m a l - d u a l g a p i n t e g r a l

Original: [Figure]
Revised: [Figure]

Revised: HEM

Revised: [Table]

Revised: setting—significantly outperforms several baselines in terms of the Time and/or Primal-dual gap integral (PD integral).

Revised: G.8. G ENERALIZE TO NON - ROOT NODES

Revised: Our learned models outperform the baselines for all nodes (both root and non-root nodes) under the one round setting and multiple rounds setting, as shown in Tables 16 and 17.

Revised: Specifically, under the one round setting with non-root cuts, our model improves the Time and Primal-dual gap integral by up to 91.29% and 29.61%, respectively.

Revised: Under the multiple rounds setting with non-root cuts, our model improves the Time and Primal-dual gap integral by up to 97.72% and 33.02%, respectively.

Revised: G. C OMPARISON WITH MORE LEARNING - BASED METHODS

Original: [Figure]
Revised: We compare HEM with AdaptiveCutsel Turner et al. and Lookahead Paulus et al. (2022) in Table 18.

Revised: The experiments demonstrate that HEM significantly outperforms the two learning-based

Original: [Figure]
Revised: [Table] [Table]

Revised: methods by a large margin in terms of the Time (up to 11.21% improvement) and Primal-dual gap integral (up to 24.36% improvement).

Revised: G. C OMPARISON WITH THE SCORE - BASED POLICY (SBP) WITH MORE POWERFUL MODELS

Revised: We have conducted the ablation study to show that the performance improvement achieved by HEM is from our novel problem formulation rather than using more powerful models.

Revised: The results in

Original: {+Table 
+}
Revised: {+Table 
+}

Revised: HEM still outperforms the score-based policy (SBP) with more powerful LSTM models, in terms of the Time (up to 80%-67.31%=12.69% improvement) and Primal-dual gap integral (up to 30.75%-22.17%=8.58% improvement).

Revised: Setups.

Revised: We implement another baseline, namely SBP with LSTM (SBP+LSTM), which parametrizes the scoring function via an LSTM encoder and a multi-layer perceptron.

Revised: In terms of model parameters, the model used by SBP with LSTM (172289) is comparable to that of HEM (212749).

Revised: Results.

Revised: HEM significantly outperforms SBP with LSTM as shown in Table 19.

Revised: The results demonstrate that HEM significantly outperforms SBP with more powerful models, suggesting that the better performance of HEM is from our novel problem formulation.

Revised: G. E XPERIMENTS WITH ADVANCED MODELS AND TRAINING METHODS

Revised: We conduct the following experiments to demonstrate that our method is applicable to advanced models and training methods.

Revised: The Attention Model Kool et al.

Revised: By replacing the pointer network with the Attention Model (HEM+AM), experiments show that HEM+AM outperforms the baselines (up to 79.61% improvement) as shown in Table 20.

Revised: The proximal policy optimization (PPO) Schulman et al.

Revised: By replacing the policy gradient method (reinforce) with the PPO (HEM+PPO), experiments show that HEM+PPO outperforms the baselines

Original: [Figure]
Revised: (up to 79.38% improvement) as shown in Table 20. [Table] [Table] G. G ENERALIZE TO OTHER SOLVERS

Revised: Our proposed methodology can well generalize to other solvers as shown in Table 21.

Revised: The results demonstrate that HEM significantly outperforms the default cut selection method in the CBC solver (Saltzman, 2002) in terms of the primal-dual gap (up to 18.67% improvement).

Revised: We do not use commercial solvers, such as Gurobi (Bixby, 2007) and (Bliek1´u et al., 2014), as the backend solver, since they do not provide interfaces for users to customize cut selection methods.

Revised: As the CBC cannot generate any cut on the dataset Maximum Independent Set, we conduct the experiments on the dataset Load balancing.

Revised: We use the primal-dual gap metric rather than the primal-dual gap integral due to the reasons as follows.

Revised: The primal-dual gap is a well-recognized metric for evaluating the solvers as well.

Revised: (2) Unlike the SCIP, the CBC does not provide interfaces for users to acquire the primal-dual gap integral.

Revised: Due to limited time, we do not implement the interface.

Revised: G. A DETAILED COMPUTATIONAL ANALYSIS OF OUR MODEL AND THE BASELINES ’ MODEL

Revised: We provide a detailed computational analysis of our proposed model and the baselines’ model in

Revised: Table 22.

Revised: We summarize the conclusions in the following.

Revised: (1) The training time of HEM and

Revised: SBP is comparable, as most of their training time is spent on interacting with solvers to collect training samples.

Revised: The model parameters of HEM (212749) and SBP with LSTM (172289) are comparable.

Revised: The inference time of HEM is longer than that of SBP and SBP with LSTM.

Revised: Nevertheless, the inference time of HEM (0.34s on average) is very low compared to the solving time (162s on average), especially on hard datasets.

Revised: The training of HEM is stable (please see

Revised: Figure 8).

Revised: G. E XPERIMENTS WITH SOME SPECIFIC STRUCTURED MODELS

Revised: We have analyzed the selected cuts on Multiple Knapsack (specific structured problems) to show that our learned policies can capture the underlying structure of the specific structured problems.

Revised: The results in Figure 9 show that our model mainly selects three kinds of cover inequalities, i.e., lifted knapsack cover inequalities (47%) (Gu et al., 1998), lifted minimal cover inequalities (43%)

Revised: (Gu et al., 1998), and flow cover inequalities (2%) (Gu et al., 1999), for solving Multiple Knapsack

Original: [Figure]
Revised: [Table] [Table] problems. Specifically, we analyze the type of cuts selected by our proposed HEM on Multiple

Revised: Knapsack, a class of problems with specific structures.

Revised: It is known that a prominent class of cut for knapsack problems is cover inequalities (Gu et al., 1998; 1999).

Revised: The results demonstrate that our learned policies can select cover inequalities for solving the knapsack problems, suggesting that our model can well capture the underlying structure of specific problems

Original: [Figure]
Revised: [Table] G.

Revised: M EASURING THE PRIMAL AND DUAL INTEGRALS

Revised: We have conducted experiments to measure the Primal Integral (PI) and Dual Integral (DI) as shown in Table 23.

Revised: The results show that the performance improvement of HEM is from both the primal and dual sides.

Revised: Specifically, we use the optimal objective values as the reference values to measure the PI/DI.

Original: 1e
Revised: However, it is time-consuming to obtain optimal solutions for all instances. We conduct the experiments on three easy datasets due to limited time. Interestingly, the results demonstrate that proper cut selection policies can improve both the PI and DI. Moreover, the results show that HEM achieves more improvement from the primal side than the dual side on Set Cover and Maximum Independent

Revised: Set, while HEM achieves more improvement from the dual side on Multiple Knapsack.

Revised: [Table]

Revised: [Table]

Revised: DI)

Revised: Improvement (%, DI)

Revised: NoCuts 39.

Revised: NA 41.

Revised: NA Default 25.

Revised: 38.25 HEM (ours) 19.

Original: [Figure]
Revised: [Figure]