Dynamic Configuration for Cutting Plane Separators via Reinforcement Learning on Incremental Graph
Authors: Mingxuan Ye
Jie Wang
Fangzhou Zhu
Zhihai Wang
Yufei Kuang
Xijun Li
Weilin Luo
Jianye HAO
Feng Wu
Venue: NeurIPS 2025 poster ()
Keywords: mixed-integer Linear Programming; cutting plane separator; reinforcement Learning; incremental graph
Primary area: machine_learning_for_sciences
TL;DR: We propose a novel Dynamic Separator Configuration (DynSep) method that models separator configuration in different rounds as a reinforcement learning task, making decisions based on an incremental triplet graph updated by iteratively added cuts.

Abstract: Cutting planes (cuts) are essential for solving mixed-integer linear programming (MILP) problems, as they tighten the feasible solution space and accelerate the solving process. Modern MILP solvers offer diverse cutting plane separators to generate cuts, enabling users to leverage their potential complementary strengths to tackle problems with different structures. Recent machine learning approaches learn to configure separators based on problem-specific features, selecting effective separators and deactivating ineffective ones to save unnecessary computing time. However, they ignore the dynamics of separator efficacy at different stages of cut generation and struggle to adapt the configurations for the evolving problems after multiple rounds of cut generation. To address this challenge, we propose a novel **dyn**amic **sep**arator configuration (**DynSep**) method that models separator configuration in different rounds as a reinforcement learning task, making decisions based on an incremental triplet graph updated by iteratively added cuts. Specifically, we tokenize the incremental subgraphs and utilize a decoder-only Transformer as our policy to autoregressively predict when to halt separation and which separators to activate at each round. Evaluated on synthetic and large-scale real-world MILP problems, DynSep speeds up average solving time by 64% on easy and medium datasets, and reduces primal-dual gap integral within the given time limit by 16% on hard datasets. Moreover, experiments demonstrate that DynSep well generalizes to MILP instances of significantly larger sizes than those seen during training.

Dynamic Configuration for Cutting Plane Separators via Reinforcement Learning on Incremental Graph
Mingxuan Ye!, Jie Wang!", Fangzhou Zhu?; Zhihai Wang', Yufei Kuang', Xijum L.i,
Weilin Jianye Hao Feng Wu!
MoE Key Laboratory of Brain-inspired Intelligent Perception and Cognition, University of Science and Technology China
Noah's Ark Lab, Huawei Shanghai Jiao Tong University *Tianjin University
fmingxuanye, y yfkuang, zhwangx]@mall wusiiiilust......m.... wstc. edu. cn, fjiewangx,fenggwee @ustc. cn
(zhufangzhou, luoweilings h Baliaanyyyhuuueeee @hhawei com edu.
Abstract
Cutting planes (cuts) are essential for solving mixed- integer linear programming (MILP) problems, as they tighten the feasible solution space and accelerate the solving process. Modern MILP solvers offer diverse cutting plane separators generate cuts, enabling users to leverage their potential complementary strengths tackle problems with different structures. Recent machine learning approaches learn to configure separators based on problem-specific features, selecting effective separators and deactivating ineffective ones save unnecessary computing time. However, they ignore the dynamics of separator efficacy at different stages of cut generation and struggle adapt the configurations for the evolving problems after multiple rounds of cut generation. To address this challenge, we propose a novel dynamic separator configuration (DynSep) method that models separator configuration in different rounds as reinforcement learning task, making decisions based on an incremental triplet graph updated by iteratively added cuts. Specifically, we tokenize the incremental subgraphs and utilize a decoder-only Transformer our policy to autoregressively predict when to halt separation and which separators to activate at each round. Evaluated synthetic and large-scale real-world MILP problems, DynSep speeds up average solving time by 64% on easy and medium datasets, and reduces primal-dual gap integral within the given time limit by 16% on hard datasets. Moreover, experiments demonstrate that DynSep well generalizes to MILP instances of significantly larger sizes than those seen during training. The code is released at wtlhtps::/// tit.. WwIRALab-USSCLLLA
Introduction
Mixed- Integer Linear Programming (MILP) problems are linear programs that involve both discrete and continuous decision variables, which have been widely used in real-world optimization tasks [13 standard MILP problem has the form:
A min xx Ax Saass,xEER b,xER".,, Z for all I).
(1)
where E R" A E R b R. I C (I.....n) n. indexes those variables constrained to be integral, and z" denotes the optimal objective value of the problem
Cooresponding Author
39th Conference on Neural Information Processing Systems (NeurIPS 2025).

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Separation Rounds at Each Nude
IPP Reloxation Regiun
Convex of Irigg Solitions
Eachalel Repion
21 H1500
Oil
+1
Configured Separators
Cut Poul
+lp b,
Conmory Chiqu J/M
Genente
cuts ou S P.
Sclect
Flow covers/ Xi) MIR 7/x
cuts 2: ove S Bg
Carer
2s (8)+1
Project 1/1x
LP the MILP subproblem
Tightened L.P relaxation
B&B Tree
Exceune the (k 1ith separation round until E++
Figure 1: Separation rounds at each node of B&B tree, At each tree node, the solver first solves the current LP relaxation. Based on the LP solution x and the current constraints, suite of configured separators- of which only the activated ones are invoked invoked-generates a pool of candidate cuts. The solver then selects the most promising cuts, adds them to the model, and re-optimizes the tightened LP relaxation. This separation cycle repeats until a preset maximum round mmax is reached.
5. which unifies Branch-and- aadnh-d (B&B) tree search with cutting-plan plane techniques into a cohesive framework. At each B&B tree node, the solver forms Linear Programming (LP) relaxation by relaxing integer constraints and solves it to yield a dual bound for the current node. To tighten the dual bound, the solver runs multiple separation rounds: find inequalities satisfied by all sneger-feasible solutions but violated by the current LP solution, add them as cutting planes, re-solve the LP, and repeat- repeat-thereby pushing the relaxation closer to the convex hull of the integer points. As shown in Fig, [ each separation round invokes several cut-generation algorithms (i.e., separators), each focuses on a particular family of inequalities (e.g.. Gomory cuts, Clique cuts). The solver then selects small, diverse set of the most effective cuts and adds them to the LP model. These separators are critical to performance because they determine the quality of candidate cuts and thus the solver's convergence behavior. Modern MILP solvers offer multiple separators with tunable configurations, enabling users to flexibly call different separators and combine their complementary strengths tackle problems with different structures.
Modern MILP solvers typically follow the Branch-and-Cun (B&C) paradigm 4
To fully exploit separators' potential for speeding up the B&C process, recent machine learning (ML) approaches [6] [ learn problem-aware configurations and disable ineffective separators to save unnecessary computing time. However, they ignore how separator efficacy changes across cut cut-generation stages, struggling to adapt configurations as the problem evolves across nodes and separation rounds. We identify two main drivers of this dynamics: (i) decaying marginal gains from successive separation rounds and (ii) interaction effects among separators. First, classic separators (e.g., Gomory, split) exhibit diminishing marginal improvements in the dual bound and introduce
potential numerical issues as rounds accumulate 18)00) Our pilot study (Fig. ( reveals that solver performance is highly sensitive to the round cutoff-more rounds do not necessarily yield better performance. Second, interactions among separators affect their joint benefit: some separator families yield strong cuts in early rounds, whereas others act as late-stage boosters; some separators provide mutual reinforcement to tighten bounds [II], while others create redundancy or dilute strength when used together [2 13 Our experiment that randomizes the activation status of separators each separation round (Fig. 2 confirms that proper round-aware configurations can deliver substantial performance gains, underscoring the need for dynamic configuration of separators.
Inspired by the analysis above, we propose DynSep, a novel dynamic separator configuration method that models separator configuration in different rounds as reinforcement learning (RL) task and jointly decides when to halt separation and which separators to activate in each round at settled B&C nodes. To avoid the overhead of re-encoding the entire MILP subproblem in each round, DynSep ingests only the incremental subgraph-the triplet graph formed by newly added cuts-at each decision-maie making step of the RL agent. Specifically, we first extract graph and node embeddings for each incremental subgraph using a Graph Convolutional Network (GCN) and tokenize these embeddings. We then feed the tokens produced at the current time step into a decoder-only Transformer, which integrates temporal context from past subgraphs and autoregressively predicts the next round's separator configurations. Furthermore, we introduce a blocked positional encoding for the Transformer that captures the temporal ordering of the incremental subgraphs while omitting the incidental order of feature tokens within each subgraph. This design ensures that DynSep performs a one-to-one mapping from each separator's features to its configuration decisions in permutation-equivarre manner, which preserves token-level alignment. Our experiments show that DynSep outperforms other state-of-the -frt---rrtttt (SOTA) configuration methods on benchmarks of both

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synthetic and scaee real-world MILP problems. Specifically, DynSep speeds up average solving time by 64% on five easy and medium datasets and reduces average primal-dual gap integral by 16% on four hard datasets, including benchmarks from MIPLIB 2017 [14] and large-scale real-world production planning problems. Moreover, our tests show that DynSep generalizes well to unseen larger MILP instances, and the visualization study shows it truncates unnecessary rounds and automatically captures some known facts of separator efficacy patterns.
2 Related Work
Learning- MILP solvers are now commonplace across industry and academia, spanning industrial solver pipelines [5], solution prediction for warm-starting [16] 17 and accelaration of core solver modules-e.g.. cut selection and branching-via machine learning [88 19 Building on this landscape, we target ML-based cut generation. Existing data-driven methods for guiding cut generation can be categorized into three classes: generating parameterized family of cuts 23) enhancing existing
separators 24 25), and configuring separators to generate compound cuts [6 7. Our work falls into the third category. Related work on ML-driven separator configuration includes L2Sep [6] and LLM4Sep (7]. L2Sep uses greedy filter to restrict the vast separator parameter space and trains an instance-wise bandit model to guide separator activation. L2Sep can adjust configurations in a few intermediate separation rounds. but requires training separate model for each round, thus limiting its scalability to arbitrary rounds and node-wise adaptation. LLM4Sep harnesses the large language model (LLM) to generate cold-start configurations for each dataset. While both methods yield notable gains, they remain confined to small or truncated parameter spaces, and ignore the dynamic efficacy and order dependencies of separators, which restricts their capacity for on-the-fly adaptation as the solver progresses.
3 Background
Branch-and-cut. In B&C, the solver builds a search tree by recursively branching on fractional variables, thus partitioning the MILP into smaller subproblems at each node. Each subproblem is
solved via its LP relaxation, yielding an optimal fractional solution that provides a lower bound (dual bound) on the objective of the subsequent subtree. To tighten this relaxation, cutting planes linear inequalities of the form ex s f-are iteratively added to the LP problem. As shown in Fig. [ these cuts strategically remove fractional solutions from the LP relaxation region without eliminating any feasible integer solutions, thus raising the dual bound at the node, Any subtree whose dual bound exceeds the best-known integer solution can be pruned; thus, cutting planes are crucial in reducing the search space and accelerating the solving process.
Reinforcement Learning. A standard MDP is formally defined as tuple M := (S,A,R, where S is the state space, A is the action space, R S x A - R is the reward function, and : x R - S is the transition function, At each time step 1, the agent observes the current state S ~ S and generates an action a, E A from its policy (s,)). This action pushes the environment towards the next state 5,+1, accompanied by scalar reward r, R(s,,a)) az). This interactive process yields an episode of sequence T := (Sp. do, Mo. $1, (1, ... ST. aT, rT), in which T means the episode ends or reaches the maximum time step. The goal of RL is find optimal policy x' that maximizes the expected cumulative rewards Bz-(x.M) E
4 Motivation
Set maximum separation round to decide when to halt. Empirical results in 2 show that altering the maximum separation round Fmax for each node causes significant performance variations. For the setcover dataset, the solving process accelerates when increasing 7max to 3, after which further rounds cause degradation. For the anonymous dataset, both solving metrics worsen monotonically as Fmax increases. Overall, these trends indicate that adding separation rounds does not universally improve performance, and the optimal value of max differs across datasets. These findings motivate us to incorporate the maximum separation round decision variable for the agent.
Set activation status decide which separator activate. In Fig. 2(b) we test 50 random configurations for each instance in several datasets and choose the best configuration specific to each
3

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Maxround (setcover) uee
Maxround (anonymous)
Performance Improv. of Configs
Solving Time
PD Integral Solving Time
10000
100%
5. 5.
Intege
75%
Salving
50%
PD (R3
2250
25%
2100
D3
D5
Datasets Picnity
Solving Time
PD Integral
Stauu
(b)
Figure 2: Motivation results. (a) Effect of varying maximum round max on solver performance for two MILP datasets. Each plot shows the average solving time (red line, left y-axis) and PD integral (blue line, right y-axis) across all instances in the dataset. (b) Performance improvement of the best configurations found by different random strategies. Datasets DI-D3 use solving time (left) as the metric, while D4-D5 use PD integral (right). The y-axis represents the relative improvement compared to the default setting. Each bar represents specific strategy to get random configurations.
instance, which illustrates the potential performance gains from different configuration strategies. The yellow bars correspond to randomly varying the activation status of all separators. Specifically, each separator's 's activation status 17 E 0,1 is randomly chosen, where 0 indicates deactivation, +1 indicates eager activation, and -1 indicates deferred activation. Thus, for each selected configuration, we randomly decide which separators more implementation details of separator configurations in Appendix B. We also randomize the separators' priorities [= which controls the execution orders of the separators in separation round. As shown by the orange har in Fig. [(b) perturbation in priority configs exhibits a minor effect on performance improvement compared to activation status (yellow bars) changes (see more results and analyses in Appendix E.I). These findings motivate us to alter the separators' activation status to decide the order of separator invocation across multiple separation rounds.
active and in which phase they are invoked. We provide
Node-wise& Round-wise Dynamic configurations. The last two bars of Fig. 2 correspond to
randomizing the activation status of all separators either each separation round (red: round-level) at nodes with regular depths (pink: node-level). Both round- and node-level random configurations show great potential in finding high-quality configurations. This indicates structural shifts exist in the evolving problems when moving to a new separation round or a new node, which leads to the shift of optimal separator configurations. These findings motivate fine-grained manner, adjusting them dynamically in each separation round at multiple nodes.
configure separators in a more
5 Method
Cutting-plane separator configuration is challenging due to the vast combinatorial configuration space and the dynamic cutting preference at different stages of cut generation. Therefore, we model the dynamic separator configuration sequential maaiinnn-maingg problem. We employ RL to explore the large action space and leverage performance feedback from the MILP solving process to guide the search for optimal configurations. In Section we present the detailed RL formulation for separator configuration. Sections 5.2 (.3 offeer an in-depth description of our proposed DynSep, detailing how model time-evolving MILP instances via dynamic triplet graphs and how to process the tokenized features using decoder-only Transformer. Section 5.4 outlines the training process for DynSep. The
overall workflow of DynSep illustrated in Figure
5.1 Reinforcement Learning Formulation
In the following, define the MDP components for dynamic separator configuration.
The state space S. Given that the entire MILP problem determines the geometric structure of the feasible solution domain-whiie which in turn provides the core information for cut generation- include the MILP problem M, to be tackled in the 1-th separation round in our state. We also embed per-separator solving feedback (e.g., cost, contribution in finding proper cuts and reducing variable domains, etc.), which guides the agent to learn the overhead and efficacy of each separator. Therefore, following the work (6), we model the above inputs as a triplet graph G CV S; with three

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Environment: the Solver
Agent: DynSep Model
State Triplet graph incremental MIL.P P instance
block
block
block Za
block 2
token block
Variable
Constraine
modes
Original Constraints As < è. visible for to
Decoder-only Transformer
Addled Aeut S. Box
GCN Eacoder
Lo F I Smiip
Actor Ty
Critte V4
visible da, (12 1)
L
Reward Py
Solving Time
Gomory
Flow cover
Incremental Pran Gup Integnl
one round
Separtic Clsin
Action tyc Maximum round & Segarator Activation Stattos
The RL framework for dynamic separator configuration. The environment (left) provides an incremental triplet graph As, of the MILP instance each time step. A GCN encoder processes As, extracting a token block Σn consisting of a graph-level embedding and K separator node embeddings. A decoder-only Transformer then processes the sequence of such token blocks autoregressively. Finally, the actor He outputs the corresponding categorical distributions of round- wise decisions: the maximum round and all separators' activation statuses. These decisions are optimized via RL loss, with rewards computed from round-level improvements in three aspects.
Figure 3:
types of nodes: variable nodes V, constraint nodes C, and separator nodes S. & denotes the edge set. Detailed node and edge feature definitions appear in Appendix [B.2
Formally, we define the state as the entire triplet graph, i.e., S = G,- However, encoding the full graph in each decision step leads expensive gradient updates during training and a growing inference time as the graph scales. To mitigate this, we instead use the incremental state As, as the input at the 1-th time step. Specifically, As, AG, captures only the updates after each cut generation round, including the newly added cuts, tightened variable bounds, and updated separator statistics. In practice, As, a triplet subgraph reflecting those changes (see the left panel of Figure 3 and we set Aso = Go using original constraints as constraint nodes.
The action space A. Inspired by the motivational results in Section 4 we define the action a, the concatenation of two decision components. First, we decide which round to stop generating cuts by a, (1) m, E 2, ... ,7)), where m, denotes the maximum number of separation rounds permitted the current node and T' denotes the predefined upper limit for separation rounds in the overall B&C process (i.e., the maximum episode length in our RL. formulation). Second, we decide which
separators to activate in early and late phases by a,²) m, II., "Txx))))) E {-1,0, 0, 1)K where 7, denotes the activation-status vector for the K built-in separators. Each entry of 1T., specifies the activation status of the i-th separator that would be setup the round r, taking values of +1 (activated early), 0 (deactivated), or - (activated late). Formally, we represent the overall action at time as:
a, concat(a," (1) a.2)) [m,, mr....... x]. E N
(2)
These configuration decisions made every separation round for nodes predefined depths.
The reward function R. To leverage the rich feedback dropped in the solving process, we define the reward as a summation of three incremental metrics: runtime penalty, relative dual bound improvement, and separator effectiveness. After the 1-th round, the agent observes
1 N
AB; B,
AT;
Thase +
AKr.
where AT, T, -T, B Ax, = K₁-1 In the first term, T, denotes the cumulative solving time up to the r-th round, Thase is the problem-specific time budget estimated under the default configuration. and N is the total number of separation rounds in an entire B&C process. In the second term, B, denotes the current dual bound obtained from the optimal solution of the current LP relaxation. Note that we normalize the incremental solving time by its estimated average per round and express the dual-bound improvement as the ratio of the increase to the previous bound, since both metries can vary greatly across MILP instances and may become excessively large, which would risk training instability. In the third term, K, signifies the aggregate gain caused by all active separators at
5

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round which is computed as the sum of: (i) the application rate of the generated cuts, (ii) numbers of domain reduction operations, (iii) numbers of node pruning operations. These three statistics provide meaningful insights into the separators" contribution to domain tightening and search-space pruning (see Appendix B.3 for more details).
The environment transition P.
The transition function maps the current state and action to the next state. That is, by performing the configuration decision a,, a round of cuts generated from the current triplet graph Sy. After incorporating all selected cuts, the solver forms an incremental triplet graph 5,+1- We denote the incremental updates as As1+1, which represents subgraph consisting of newly added cut nodes, updated variable and separator nodes, and newly connected edges.
The terminal state. At the first separation round of each node, the agent provides decision of maximum round mo. Then the solver environment terminates cut generation after mo rounds or other built-in stopping conditions are met (see Appendix [3.4 for more details).
5.2 Tokenize Incremental subgraphs
We extract latent embeddings for both the entire graph and its separator nodes using the encoder network & introduced by [6]. The encoder & involves a Graph Convolutional Network (GCN) [26), an attention block on the hidden embeddings of the separator nodes [27]. and a global pooling layer that produces single representation for the input graph. Full details of the pessaae-passiigg operations are provided in Appendix [ here, we focus on the tokenization procedure that is unique to our approach.
At each time step r, feeding the incremetal subgraph As, = AG, into the encoder & yields a graph feature hg, Rdd and K separator node features (t)),.))...... .hh,....,,,.... Rd. We treat each feature vector as a token, and then we map the incremental subgraph AG, into a stacked token block:
2 =
[ hg.(t.1) ... ".,,,,))) &
(4)
5.3 Decision-Maker: Decoder-only Transformer
The token block z, is an embedding of the incremental state As, containing incremental contexts and separator statistics updated in the last separation round. To recover the entire information of state for informed decision-makinn making, we aggregate the information of all history incremental states (s;(0 i via a decoder-only Transformer T.
Decision-making in an autoregressive fashion. At each round r, we assemble the history of token blocks as ordered sequence W,, = (Zo. 21....,2 Zy) We input w,, into the Transformer T and train T to predict the corresponding configuration actions (ag. dj...., a,) in an autoregressive fashion, where a; [m,, 1...: K for = 0 r. To optimize decision quality in online RL manner, replace the common prediction error loss in Transformer models with the RI objective that maximizes the cumulative return the generated actions. During inference, we feed the historical sequence w,, to T and generate the next action a,, like the next-token generation in language modeling.
Blocked positional encoding. We introduce blocked positional encoding (PE), enabling the Transformer model to recognize and capture the temporal ordering among token blocks, while omitting any ordering among tokens within each block. Formally, define the blocked PE of one episode as
P= [P] PT
PT] E ((T+))(K+)xd
(5)
where P = [pos(i) pos(i) ... T E (K++))xd pos(i) E Rd. That is, the blocked PE assigns the same positional encoding pis(i) to all K + I tokens in ž₁. Then the input to T becomes w, = (zo + Po.....2. 2, + P,). This design mirrors relative-position schemes that bias attention at the block level while leaving intra-block tokens interchangeable. Similarly, we a blocked causal mask that forbids attention from block i to any future block > i. At inference time, we take the last K + hidden embeddings
embedding block to predict the next action.
Permutation equivariance inside each block. Note that the token block 2, in Eq. and the action in Eq. (2) are consistent in both structure and contextual meaning. Thus, we utilize the embeddings of K + tokens 2, after passing the Transformer T to obtain the + probability distributions of K configuration components in action a,, respectively. We formalize the one-to-one decision

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mapping as follows:
block-level: (ao..... T ((z0 + Po...... ......... Pi))
(6)
token-level: a = It.1........................,... n..k] = hs..i. (.. (T) +
(7)
In Eq. 77. the first token fg.. of each block zi informs the global mode of subgraph As; to decide which round m; to stop. The subsequent K tokens hs.(6..)...... (...))........(...))) guide separator configs Tr....... ................................ Mi, x. We visualize the data flow of block-level and token-level decision mapping in the right panel of Fig. 3 Such a one-to-one mapping encourages the agent to make context-awand decisions and is guaranteed by the permutation equivariance of the self-attention mechanism when the input tokens share the same position embedding. We formulate such permutation equivariance as follows and provide the corresponding proof in Appendix
The decoder-onts Transformer T equipped with the blocked positional encoding P. permuttation equivariant inside each token block. Formally, for any input X and any block-wise permutation matrix II, it holds that (IIX+ + P) =
Proposition 1.
5.4 Training
We utilize the Proximal Policy Optimization (PPO) algorithm 28 to train our decision network, as PPO strikes a balance between ease of implementation and computational efficiency. PPO inherits from the famous actor-critic framework in which critic (s(s) estimates the state value function, while an actor xa determines the policy. We instantiate two separate decoder-only Transformers for the actor and critic network, respectively. each sharing the architecture defined in Section 553] As shown in the right panel of Figure 3 Vi only takes the first token (the graph feature) of cach block z from the Transformer's final hidden embedding and then feeds it into a linear head predict the state value. The actor concatenates the Transformer's s final token embeddings and feeds them into joint linear head to produce the probability distributions of all configuration decisions. We provide the algorithm pseudocode in Appendix D.I and the architecture hyperparameters in Appendix D.2
6 Experiments
6.1 Experimental Setup
Setup. We use SCIP 8.0.0 29 as the backend solver, which is a state-of-the-art open source solver widely adopted in the research of ML for combinatorial optimization (ML4CO) 30
31
I8
32].
To maintain fair comparison and reproducibility, we retain all the other SCIP parameters as default in baselines and our method. All the SCIP solver's advanced features, such as presolve and heuristics, are open, which ensures that our setup is consistent with the practice setting. Evaluation on each instance is limited to a 300-second solving time. To balance performance and computational cost, approach configures separators at every tenth depth level in the B&B tree by setting the separator frequency in SCIP as 10. We set the maximum separation round at each node as T 5. Configuration is applied to 22 separators built into SCIP; more descriptions of separators are provided More details about the ML. setup and hardware specifications are provided in
in Appendix B.5 Appendix D
Benchmarks. We evaluate our approach on nine publicly available NP-hard MILP problem benchmarks from the prior work [B3], covering three classical synthetic MILP problems and six challenging MILP problems from diverse application areas. The nine problem benchmarks are divided into three categories (easy, medium, and hard) according to the difficulty of solving them using the SCIP 8.0.0 solver. (1) Easy datasets: three widely used synthetic MILP problem benchmarks: Set Covering [34], Maximum Independent Set (MIS) [35, and Multiple Knapsack (36). (2) Medium datasets: CORLAT [37] and MIK [38), which are widely used benchmarks for evaluating MILP solvers (39 40 (3) Hard datasets include two datasets from the ML4CO NeurlPS 2021
Competition 41]: the Load Balancing problem inspired by real-life applications of large-scale systems, and the Anonymous problem inspired by large-scale industrial application. Moreover, hard datasets contain MIPLIB mixed neos and MIPLIB mixed supportcase, two subsets of the benchmark
MIPLIB 2017 [14.
Baselines. Our baselines include four human desanndd separator configuration rules and three based methods. Human-designed rules include (1) Nocuts: pure B&B without adding any
7

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Table I: Comparative evaluation on easy, medium, and hard datasets. Best performance is in bold, with the greatest improvement (Improv.) both bolded and underlined. Sizes of nine benchmarks are in parentheses, with n and m representing the average numbers of variables and constraints, respectively. The values report the mean (standard deviation) of time and PD integral metrics.
Easy Set Counning - 1000, in 500)
Fasy: Mas Independent Set (n 500, m 1953)
Easy: Mulliple Knapsekk (o 720, v 72)
Method
Timeis)
Improv 7
(time, %)
PD integral
Time(s)
Impros. T (lime" (6)
PD integral L
Timels) 1
Impros: [tiee %)3
PD iniegral
NoCuts
7.45 (5.87)
NA
191 26 (35.59)
15.32
NA
146.4 (26. 99)
13:84 (28,
NA
25.21 (26.6)
Datault
5.24 5.24(1.7))
20.66
(36.8)))
30.4 (11488.00 (0))
-98.43
289.5 (103.8)
2.01 (1.82)
85.48
(10.9)
Search(50)
1.7 (0.44)
77.18
36.77 (8.48)
4,42
71.15
23.28 (19,02)
4.32 (5.52)
70.25
13,38 (7.36)
Pruse
0.00
(45.97)
67.17
30.93
0.64
95.38
9:89
6.56 (4.25)
13.95
(39.08)
5,42 86)
64.62
34.21 (25.07)
7:45 (2.07)
46.17
12.99 (8.60)
1.28ep(R2)
7.35 (4.58)
5.56
65.01
(22.9)
33
(8.72)
LLM
11.73 (12.09)
-5.45
(91,14)
(4.19)
66.51
27.18 (20.17)
(5:51)
(8.9)
DynSep (Curs)
1.51 (0.27)
79.73
3388 (9.34)
(0.20)
96.54
9,66 (2.40)
0.52 (0.24)
96.24
9.71 (5.39)
Mediam Codat (e 466, 485)
Medium: MIK (s 413, = 146)
Hard: Anorgmoss (e 17881, is = 49600)
Mathod
Tincte)
Impeox.
iniegaal
Timote)
Improw
PD integaa
Timzto 1
integral
Improw:
NoCuss
74 (122.28)
NA
2687 68 (6209 46)
100.25 (115:97)
NA
867.85 85 (350 76)
250 37 (75.71)
21117. 12 (9234 D1)
NA
Detuih
111 (352.) 1M)
-9.41
10573, 14 (13070 46)
16.55 (15.06)
91.25
82.80 (56, 24)
258 (4 199
27000 38 (1892.N) S)
-2K.19
Scaech150)
(97.19
25.34
2910.77 (6545.5) St
24.99 (20.56)
86.87
99.27 (55. 83)
270.68 (65. 52)
24005 68 (5007.57 (77
-13.79
Prane
8900 (M255/1255807)
-19.35
2015.71 (5816
300.01 00.00
-37.67
2239 28 (1023 80)
241.75 (10061))
17 (9363.5)
18.05
L25ep(R1)
(12
3124,07
(17,64)
61
53
40)
19.64
(5027
11.13
240.54
18850
(10052.83)
64.03 (110.53)
2921.73
21)
(17.76)
90.57
264.57 (34.
25384.68 (8100.5)
-20.3
DynSop (Damn)
21,96 (18.90)
2235.42 (8H6845))
10.99
94.22
130 (64.21)
15656.7 (899614)
25,96
Hud Load Bulascita ie 61000 64304)
Hadd MIPLIB nixel 8209 (1 6904, xn 5660)
Had: MIPLIB uisel supporase 19766,
199100)
Method
Timets
PD interrd
(PD
Tiees
anogral 1
(PD
Impron
Tisets)
inegral 1
Insprox:
NoCais
309.31 11 (0,02)
15993 26 (140.663
NA
275.04 04 143 23)
14618
(1211463)))
N/
181 (1 (130 25)
12M59:5 (T0506.47)
NA
Delant
500. 44 (0022
1S197. 19 (936, No
.862
281-8 129 29)
18800.5 (9386 15)
-28 56
244. 15 (105.8) $3
2156
-6633
Search150)
300:04 (0.05)
3783.52 52 (448,000
94.35
234.33
15819 (969447
-635
133.36 (131 35)
10241.17
(10754.99) 59)
20:05
L25cy(R1>
300.82 (8,85)
10548.89 29 (4474 88)
30.11
242.83 (59 (2)
16583 49 (11911118888 LE)
25,97
162 (1.188113)) (1.38 25)
11518.55
(31756.55)
12.67
L2Scy(R2)
300,03 10)
10800 13 (4348800 969
25.05
242.50 198 (0)
13589109 (12116.35)
4.31
166 (134-71) 71)
11489:15
(11841115)
11:35
900-00 (0.00)
47: (100
275.14
14109 (13/06 L8)
3.48
156 (100 581
22815
(10274331
-34.5
DysSep (Chm))
3720.26
cuts; (2) Default: default separator configs used in SCIP 8.0.0; (3) Search(p): randomly samples p configurations then applies one with the best performance on the validation set; (4) Prune: deactivates separators with no contribution during the evaluation on the validation set. For learning- based methods, we compare: (1)L22 L2Sep [6] that learns the configuration only at the first round: (2) 1.2)ep(22): 1.2Sep that learns the configuration only at the first&second rounds; (3) LLM4Sep i : configure separators via LLM. Please see Appendix D.4 for details of these baselines.
Evaluation Metric. We use two widely used evaluation metries: the average solving time (Time, lower is better), and the average primal-dual gap integral (PD integral, lower is better). We assess different configuration methods by measuring the relative improvement compared to the NoCuts
baseline: 5(-) = Exex ChtriicNoCCus--Meeeee
where X is given dataset, and Metric(-) reflects
(Noris
the performance of method under the given metric. For easy and medium datasets, we use solving time the evaluation metric, the solver augmented by DynSep can solve all these instances to optimality (i.e., the average of primal-dual gap is zero, see detailed results in Appendix E.2) within the given time limit. For hard datasets where optimality is not always achieved within the time limit, we adopt the PD integral [33 42 41 43) to quantify the cumulative distance between the primal and
dual bounds over the solving time, where the primal-dual distance (gap) reflects the solution quality, and the integral over time exhibits the solver's efficiency of converging to the optimal solution.
6.2 Experimental Evaluation
Experiment 1: Comparative Experiments. T shows that DynSep significantly outperforms the baselines in both solving time and PD integral. On easy and medium datasets, DynSep accelerates average solving times by 64% compared to the SOTA learning-based baseline. DynSep achieves lower PD integrals than baselines on four out of five easy and medium benchmarks, demonstrating faster convergence enabled by our dynamic configuration method. For MIK, the relatively large PD integral indicates slower convergence in the early stage, but accelerated convergence in the later stages. For CORLAT, We emphasize that DynSep is the only configuration method that solves all instances to optimality with zero primal-dual gaps (see results Appendix F.2)., highlighting the effectiveness of DynSep for fine-gre -ra-grr separator configurations. On hard datasets, DynSep reduces the average PD
8

---

Table 2: Ablation study comparing five variants of DynSep three datasets.
Easy: Mulliple Knnpeack (n- 730, 72)
Medium Codlat 46f, 486)
Hard MEPLIB mised note (n 6058,
566t)
Methed
Timeto 1
Ingrex: (tme, (6)
PD integral
Time( 1
Impros. (time 9g)
PD integnl
Timejs) 1
PD integnal 1
Improv. (PD Inc. (6)
NoCuis
1384 (28.79)
NA 85.48
25.21 (26.6)
74,66 (122,22) 23)
NA
2087.68
(6209) 463
275.06 06 (43.23)
14618,53 (12214.63)
NA 26.56
Defaut
201 (1.52)
(10.9)
111.55 (132.19)
49.41
10573.14
282.98 (99
18500.5 (9386. 15)
w/o MarR
(0:46)
95:15
(5:35)
(5% 99)
(5434
(6%.0)
4129.4 9)
(0.53)
95.38
(5.244
(61.58)
(6 16146-00
245 04 (95
9448
(1.23)
10:00 8b)
23,06 (50.97)
2635,47
(5086.34)
9235:22
110:31 (131
112587.244
30000
17330.56
(1.12
(1283367)
DynSipp (Cum)
(0.34))
238.19
(12413.9)
integral by 16% within the 300-second time limit. DynSep significantly accelerates solving time on challenging MIPLIB datasets and shows comparable time on the other two hard datasets. In contrast, other ML-bbaeed baselines improve efficiency on some datasets but struggle to maintain performance across all datasets from various scenarios. Search(50) performs well on some instances, but requires evaluating a large number of configurations in advance and suffers from instability across different problem types. We also evaluate the inference latency and memory overhead of our configuration policy in Appendix E.5 and observe that while inference time increases with instance size, it does not compromise the overall efficiency gains achieved by our method. For completeness, we also report two complementary studies in the appendix: evaluation on additional MIPLIB 2017 datasets (Appendix E.3) and an extension of DynSep to broader solver hyperparameters (Appendix E.4p. both showing trends consistent with our main results.
Experiment 2: Ablation Study. We present ablation studies Multiple Knapsack, CORLAT, and MIPLIB mixed neos, representing easy, medium, and hard datasets. We provide completed results on nine benchmarks in Appendix [E.61] To understand the contribution of each component in DynSep, we evaluate the following five variants: (1) w/o MaxR: DynSep without the decision-making on the maximum separation round m, at each node. (2) w/o Ty:Dyn using Long Short-Term Memory(LSTM) architecture as the decison model of DynSep, instead of the encoder-only Transformer
designed in Section 5.33 (3) w/o DynG: DynSep without the design of dynamic graph, instead inputting the entire triplet graph at each separation round. (4)w/o DynG&TF: DynSep without the design of dynamic graph, while replacing the Transformer with an LSTM model. (5) w/o BlockPE: DynSep without the design of blocked positional encoding, instead using a standard token-level positional encoding that assigns a unique position to each token irrespective of block boundaries. Table shows DynSep overall outperforms the five variants, with comparable results in the few remaining settings, indicating that all components are essential to cope with evolving structures in challenging MILPs. We further report ablations on the encoder architecture and on hyperparameter sensitivity e.g.. separator frequency and maximum separation round in Appendices E.6.2 and E.6.3
Experiment 3: Generaization Tests. We evaluate the ability of DynSep to generalize across different sizes of MILPs. Fol-
Table 3: Evaluate the generalization ability of DynSep on MIS
Sen Courring - 1000 1000, 2x)
Sei Courring 1000
2000 4x)
Mishaal
Timele)
clime 95
Impecs:
PD integral
Tinaiol 1
Imprrs: (time 90)
PD integna
NuCuts
78.91 (76.53)
283.27 (52.80) 26)
31.9 (101334)
lowing prior works (30 33), we test the generalization ability of DynSep on Maximum Inde-
Defasit
(3.55)
19:74 (5.06)
93.01
686
(10N7.00
107.96 (90.55)
36,69
826.35 (650.27)
295,01
4505.02 (1302.02)
LEScp R1)
105.48 (92.68)
-33.67
803.32 (656.39)
(31.4)
-1.66
4728.1 (1325
L.5cp R2)
115.43 (95.5)
.47.5
905.95 (694.71)
203.77 96)
-.57
4481.42(1377.6))
pendent Set (MIS), as we can artificially generate instances with arbitrary sizes for these synthetic MILP problems. we test 4x and 9x larger instances of MIS. Table 3 (right) shows that DynSep significantly out-
140.20 16. (105-45)
-82.18
1115.22 (167.14)
795 (20:14) 14)
-468
5881 (100) ST)
DynSep (Curs)
4J03 (9.64)
94,89
104.02 (20.19)
19.05 (26.92)
93.25
629.28 (1106.30)
Inikperddan
1000, 3946,
Mas Indeperades
1500,
5940,
Mechdd
Tisaia
Impros: 1
(limee 36)
PD insegna 1
Tincio1 1
Impros: (tine, 15)
PD istegna
NeCuts
193.53 (95.78)
NA.
1056 83 (544.51)
200.01 (0.01)
NA.
2226.77 72 (370.66)
Dataile
(66.01)
S491
16 6517. 14)
19 (31 76)
40.44
1762.96 (887 (33)
Scardh/30)
1S1.51 (98.42)
23.51
23)
209 17)
1251 (931433.300 (00
Prute
(05,33(56440
45.58
396.8 (318.97)
292.94 (26.4))
2.30
1312.04 (343.31)
performs the baselines in terms
L.Sep/RR2
125qpHR3)
144.67 (94.46)
(9)
29.00
26.01
546.58 (370.45)
$30,09 (367.16)
299 (6.
299.9 (5
0.17
0.22
1508.9 (354.98)
1494,7 (346,61)
of the time and the PD integral, demonstrating the superiority of DynSep in generalization capa-
DynSop (Cun)
L.M44Sppasc
60.68
5.47 (19.09)
(42.08)
68.97
97.20
123. 98 (143.20)
36.75 (71.49)
264.44 06)
26,66 (77.49)
11.56
91.11
151.39(3364.))
(337.13
bility. We further evaluate cross-domain and general-to-speeiffi generalization; see Appendix B.7
Experiment 4: Interpretability Analysis. We visualize our learned separator configurations in Fig plotting the averaged value of 22 separators' activation status at T = 5 separation rounds. Our findings

---

Sepa. Config at Each Sepa. Round [mik]
Separation round
.1.00
1.00
Activation Status
0.90.:1
0.30
0.00
0.82
0.82
1.00
0.00
0.00
0.08
0.18
0.09
0.00
0:82
1.00
0.91
0.00
0.82
0.82
1.00
0.09
1.00
0.82
-0.5
-1.00
0.00
1.00
0.00
1.00
0.82
1.00
0.53
4.227
1.00
0.82
0.0
1.00
1.00
0.82
1.00
1.00
-0.
-1.00
0.00
0.00
1.00
0.00
1.00
0.00
0.00
0.00
1.00
1.00
0.00
0.45
3.82
3.00
0.00
0.,82
11.00
11000
0.91
1.00
1.00
contrrate
dijunctiee
minne
mekip
Intermints
concensre)
pauge
Implint
gemory
grmip
streiging
apprepation
chique
zeroholi
eccato
astrrycle
Rowrow
rapidfeaming
Figure 4: Separator configs at each separation round for MIK benchmark. The x-axis lists the 22 separators, while the y-axis shows their average activation status over every node and instance in MIK. Green hue indicates a tendency to apply the separator before the constraint handler, red huc means after the handler, and pale color means the separator generally disabled.
include: (1) Fig. 4 shows that DynSep consistently activates the CMIR separator (2nd-rightmost col.) in MIK problems, which aligns with the known facts that mixed-integer rounding (MIR) cuts are particularly effective for knapsack-type structures [38]. (2) For easier benchmarks such as MIS and Set Covering, the policy uniformly reduces the maximum number of separation rounds to mmax at every node (See Figs. MandB in Appendix (E.), demonstrating that our learned decision on maximum round effectively prunes unnecessary cutting rounds on simple problems. (3) The heatmap reveals that the separator configuration not static but varies dynamically across separation rounds (shown along the y-axis), suggesting that the model is timing the application of various separators to coincide with the stage of cut generation. Appendix E.8 provides more results and analyses on other datasets.
7 Conclusion
This work proposes a novel dynamic separator configuration (DynSep) method that formulates round-wise separator activation and stopping a tokenized incremental subgraph and uses a decoder-only Transformer to autoregressively predict separator activations and termination. DynSep significantly improves MILP solving efficiency on both synthetic and large-scale real-world benchmarks and generalizes to larger unseen instances. The current implementation of DynSep lacks a lightweight decision model to retain historical information across the global B&B tree. Future research should focus on developing such models to enable incremental decision-making based the updated graphs.
RL. task. At each decision step, DynSep processes
8
Acknowledgements
The authors would like to thank the anonymous reviewers for their insightful comments. This work was supported in part by the National Key R&D Program of China under contract 2022ZD011980 19801, National Nature Science Foundations of China grants U23A20388, 62021001 and 601B1011. work was supported in part by Huawei as well.
This
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Table of Contents for Appendix
A Proof of Permutation Equivalence
23
Formulation for Transtormer
23
A.2
Proof of Proposition]]
23
Implementation Details of Separator Configuration in SCIP
24
B.1
Frequency and Priority Setup SCIP
24
B.2
Input Features of the Triplet Graph
25
B.
Separator Statistics
25
B.4
Stopping Conditions for a Separation Loop
27
B.5
Separators Built SCIP
27
C
Network Details
29
The Encoder Network
29
D Implementation Details of our algorithm
29
D.I
Algorithm Pseudocode of DynSep
29
D.2 Hyperparameters
................................
29
D.3 Hardware Specification
31
D.4 Baselines
31
Additional Results
31
E.1
Motivation Results on effects of difterent configurations
31
Evaluation Results Other Metrics
................................
32
Evaluation on Additional MIPLIB Datasets
33
E,4
Extended DynSep to Broader Solver Hyperparameters
34
Overhead Evaluation
35
................................
E.5.1
Latency Policy Inference
35
E.5.2
Memory Overhead
35
E,6
Ablation Study
36
E.6.1
Module Ablation Analysis Other Six Benchmarks
................................
36
E,6.2
Encoder Architecture Ablation
36
E.6.3
Hyperparameter Sensitivity Analysis
36
E.7
Generalization Study
37
E.7.1
Cross-Domain Generalization Test
7.7.2
General-to-SSeeiffi Generalization Test
37
E.8
Visualization of Separator Configurations Nine Benchmarks
38
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A Proof of Permutation Equivalence
A.1 Formulation for Transformer
The standard Transformer architecture comprises two main components: the multi-head self-attention layers (MHA) and the position-wise feed-forward network (FFN). In the following part, will
briefly introduce these blocks. We represent an input sequence as X = .XI.....NN) Rxxd where X; is the hidden embedding for the token i,nnd is the dimension of the embeddings. The MHA module projects X to a triplet (Q, K, V), as follows.
Qx XWe. Kx XWR. Vx = -XWv.
Qx X
Attention(X) X) softmax
vdx Vx.
where We E Raxdk, Wx E Raxdk, Wy Rdxdy are learnable weights, with dx dv # Overall, H such projections performed, resulting in (Qx (h) v(l)) for h H.The
self-attention operation is then applied to each triplet:
(h) X
v(h)
head, = softmax
(8)
Vdx
x
(H)(H) concat(head,, .... headæ) ) Wo
(9)
where Wo Rdxd is learnable weight matrix. The output of the MHA module is then passed through feed-forward network layer, followed by a residual connection and layer normalization (LN). Finally, the output of the /-th layer H' is computed follows:
H' LN(H MHA(H' )).
(10)
H' LN + FFN(H
(11)
To remain consistent with the notation the common Transformer study, a small subset of definitions in this appendix overlaps with those in the main text. We claim that these symbol definitions valid only within this chapter.
A.2 Proof of Proposition
We now present the full proof of Proposition defined by
Recalling that the blocked positional encoding is
p= [P] PT
PTJ ((T+I)(KiI
(12)
where [pos(i) pos(i) p (K+1)xd pos(i) Rd, so that each of the K tokens in block i shares the same vector pos(i) E Rd.
Proposition. The decoder-only Transformer equipped with the blocked positional encoding P. is permutation equivariant inside each token block. Formally, for any input X and any block-wise permutation matrix II, holds that (IIX- + IX + P).
Proof. Because forwadd layer- normmaiziazi,,,n,, and residual addition each of them either applies elementwise or row-wise operations, or multiplies on the right by a learnable weight matrix, they automatically commute with any ow-perrutattion of their input. Hence, to prove that the full model equivariant under block-wise token permutations, suffices show that each self-attention module satisfies At Attention(IIX P) II Attention(X X + P).
We define the block-wise permutation matrix II permutation matrix that operates independent permutations within each block, which is formulated as follows:
Th
11
II
I
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where II, {0, 1)((+))x((+)) is an arbitary permutation matrix, with the condition that each row and each column contains exactly one entry of and the rest are O. Denote the input by:
x [z0 z] 2 E (K where ž [hipo h1 hi,x' E R (K+1)xd
Apply II yields X = πIX [[oo]] 1 ... 1 Because each block Pi in our positional encoding consists identical rows, permitting these rows does no change, i.e., II,P; P, for any i. Hence, we have that
IIX P = IIX + NP = I1(X + P).
(13)
Next, after performing the blocked positional encoding, have the attention of X follows:
Attention X P) stftmax (IIX P (IIX P)"
P)W+
(14)
vdx
Since softmax is applied row-wise, it can be viewed as left-multiplying the input by matrix. Thus, by the associativity of matrix multiplication, we can freely regroup the products of the matrices in the above formula (14 First, deduce that
Σ (12 P + P,)
IIX+P) (IIX T
1=0
Σ
II I1 T
1-0
Σ
7 zi
P; P
(15)
I=0
(z) + (z;++
PX+
(16)
where the deduction of equation
15 utilizes the property that II and 11 P.II.
Pi.
Therefore, substituting equations
16 and 13
into
14. we have
(X + PIW WT (X
Attention( x P) = II softmax
(X+ P)Wv
(17)
Vdk
QxK Vdx
II softmax
Vx
(18)
II Attention(X + P).
(19)
Finally, each subsequent layer (feed-forward, layer norm, residual connections) also commutes with block-wise permutation, so the entire model satisfies the block-wise permutation equivariance:
T(IX + P) (X
as required.
B Implementation Details of Separator Configuration in SCIP
B.1 Frequency and Priority Setup in SCIP
In the SCIP solver, we adjust the activation status configuration by two hyperparameters: the frequency fi and the priority qi- We provide the detailed description of these two configuration parameters as follows.
SEPA_FREQ f:: The frequency parameter determines at which nodes in the branch-and-boune tree separator invoked. Specifically, setting the fi = disables the separator entirely while fi activates the separator in any separation round of any tree node. Any positive activates the
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separator at every node whose depth is multiple of ft. We set fi = 10 for all active separators in our paper.
SEPA_ pj: The priority parameter dictates the order in which separators are executed during separation round a node. In every separation round, all separators with p; O are executed first in descending order of Pt. then constraint handlers are applied, and finally separators with < O run in descending order Pi. By convention, separators implementing fast, high-impact cuts have large non-negative priorities so that their cuts are added early, thus strengthening the LP relaxation sooner. In contrast, more expensive or specialized separators are given negative priorities. Hence, they run later or only if no earlier cuts were found, thereby avoiding unnecessary overhead at the start of each node's ' separation phase. This division into early (p: 2 0) versus late (pr < 0) activation phases directly influences the quality of intermediate bounds: running aggressive separators early can dramatically tighten the relaxation and reduce the number of branch-and-bound nodes, while deferring or disabling them can save CPU time when their benefit is marginal at that stage.
Activation Status To encapsulate the activation configuration, we define an activation status variable for each separator. In detail, = O means that we set the frequency of separator fi = 0 and thus let it never run in all B&B tree nodes. Fii +1 means that we set fi 0, but the priority of the separator P; >= O, executing it before the constraint handler [44] 'h -1 means that we set fi > O and Pt 0, which means the separator is activated but executed after the constraint handler. As shown by the orange bar in Fig. 2(b) of the main text, perturbation in priority configs exhibits a minor effect on performance improvement compared activation status (yellow bars) changes This phenomenon arises because activation status dictates the order of separators across rounds, whereas priority affects only their relative order within a round; since the LP is not re-solved until the end of a round, reordering separators within the same round has little effect on overall performance.
B.2 Input Features of the Triplet Graph
Node Features Each node aype-variables, constraints, and separators-is is characterized by a set of features that encapsulate their properties and roles within the optimization process. The input features for variable nodes V. constraint nodes C, and separator nodes are list in Table 4 incorporate two graph-level input features-the dual degeneracy rate and the Furthermore, we ratio. Concretely, the dual degeneracy rate is the fraction of nonbasic variables having zero reduced cost, and the suriable-to-constriie ratio is the number of unfixed variables relative to the LP basis size. In practice, we first apply an additive (sum) pooling over the GCN encoder's node representations to obtain a single aggregated node feature vector. We then append (concatenate) the two scalar metrics to this pooled embedding. The combined vector is passed through a multilayer perceptron, yielding a unified graph embedding that fuses the local structural information.
Edge Features Like the bipartite graph modeling for common MIIP problem, construct edges between variable and constraint nodes such that a variable node Vi is connected to a constraint node C2 if the variable appears the constraint with the weight corresponds to the coefficient Aij # O, and we set the value of the edge as Aij.
B.3 Separator Statistics
In SCIP's stlver-statistic display, each separator reports several key metrics that provide insights into its performance and impact during the solving process. We use these metrics as input features of separator nodes and immediate reward signals at each time step. These metrics include:
Cut Application Rate. This rate measures the effectiveness of a separator's generated cuts by calculating the ratio of cuts applied to the LP relaxation to the total number of cuts found. The application rate is computed as:
Cut Application Rate Number of Cuts Applied
(20)
Number of Cuts Found
A higher application rate suggests that the separator frequently generates cuts deemed valuable and effective by SCIP's internal filtering mechanisms, leading to their inclusion in the LP relaxation. Conversely, lower rate may indicate that many of the separator's 's cuts are redundant less impactful.
Domain Reductions (DomReds) Separators can also perform domain reductions by tightening variable bounds through their logic (for example, by deducing x; u or xy 2 from cut coefficients).
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Table 4: Description of input features for variable, constraint, separator nodes, and the entire graph.
Type
Feature
Setting
type has _l lb has_uh ub
Variable's type: binary, integer, implicit integer, continuous (one-hot).
If the variable has an infinite lower bound.
The variable's lower bound, set 0 if is infinite.
If the variable has infinite upper bound.
The variable's upper bound, set if is infinite.
basestat coef_norm
Simplex basis status: lower, basic, upper, zero (one-hot) Objective coefficient, normalized by objective norm
redcost_nome norm
Variable's reduced cost divided by the objective norm, indicating how much the objective would worsen per unit increase at zero slack
Vars
age
The number of consecutive LP iterations in which the variable stayed at zero in the basis.
solval
The primal LP solution value of the variable. The fractional part of 'solval'.
solfrac
sol is at lb
"solval' equals the lower bound within numerical tolerance If solval equals the upper bound within numerical tolerance Index of the current separation round
sol is at ub round num
origin wrigin_type
Which mechanism generated this row: unspecified, constraint handler, constraint, separator, reoptimization (one-hot).
origin_sery sepa
The separator name that produced this row.
basestat
The row's basis status in the LP solution: basic, lower, upper (one-hot). Unshifted side normalized by row norm.
bias dualsol is at Ihs is at_ rhs norm_nnz
Dual LP solution of row, normalized by row and objective norm. If the row value equals the left-hand side.
If the row value equals the right-hand side.
Cons
Number of nonzero coefficients in the row, normalized by the total number of LP variables.
age
The count of successive LP iterations for which the row has stayed nonbasic at zero.
int_ int_support
Integral support score of row.
is s__interral
Activity of the row always integral in feasible solution. Row is removable from the L.P
is removableeee
is is_in_lp
Row is member of current LP. Index of the current separation round
round _um
type
Type of the separator (one-hot). Execution time consumed by the separator.
time
calls
Number of times that the separator has been invoked.
Sepas
cuts
Number of cuts generated by this separator.
cutoffss
Number of infeasibility detections (cutoffs) found by the separator.
domreds
Number of domain reductions found by the separator. Number of cuts from the separator that were applied to the LP relaxation.
applied
dual rate
rate
The proportion of nonbasic variables with reduced cost zero.
Graph
ear_con__ut cot_rattio
The ratio of unfixed variables to the size of the LP basis.
Each time separator callback successfully reduces a variable's domain, the DomReds counter is incremented. This statistic captures the total number of such hound-tightening operations executed by that separator during the solve. A higher DomReds count signifies that the separator contributes significantly to shrinking feasible regions, which can indirectly improve future LP relaxations and cut generation efficiency.
Cutoffs. Within each separation round, when separator generates one or more cuts that render the LP infeasible or whose bound surpasses the current best primal solution, SCIP immediately prunes that node (i.e., cuts it off) without further processing. The Cutoffs statistic for separator is simply the total count of these pruned nodes attributable to its cuts. In practice, a high Cutoffs value indicates
26

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that the separator is highly effective at early fathoming of unpromising subproblems, potentially tree.
reducing the size the
B.4 Stopping Conditions for a Separation Loop
In SCIP, the separation process at node is conducted in iterative rounds, where each round involves generating cutting planes to refine the LP relaxation. The separation loop at a node terminates when any of the following conditions are met:
Maximum Number of Rounds Reached: user-defined limit on the number separation rounds per node is enforced to prevent excessive computation. For experimental stability, we use only the maxround m, decision from the first separation round at each node to set the maximum number of subsequent separation rounds at the current node.
Stalling Criterion Triggered: If consecutive separation rounds fail to yield improvements in the objective bound or integrality, the process is considered to have stalled, prompting termination.
Relative Bound Distance Exceeded: Separation is halted if the relative distance between the current node's s dual bound and the global primal bound surpasses a predefined threshold, indicating diminishing returns from further separation.
No Further Separation Requested: all separators and constraint handlers indicate that no additional separation is necessary (i.e., none return status requesting another round), the loop concludes.
These stopping criteria ensure a balance between the thoroughness of the separation process and computational efficiency, preventing unnecessary iterations that offer minimal benefit to the overall solution process.
B.5 Separators Built In SCIP
We consider 22 separators in our configuration task. We provide the detailed description of these separators follow.
closecuts. Close cuts are a type of cutting plane that focuses on generating cuts that are "close" to the current fractional solution. These cuts are designed to tighten the feasible region by targeting solutions that are near the boundary of the current relaxation. The idea improve the quality of the LP relaxation by adding cuts that are particularly relevant to the current solution.
are a class of cutting planes used in mixed-integer programming. particularly based on the concept of disjunctions. These cuts are derived from disjunctive argument that partitions the solution space into different disjunctive sets. By analyzing the infeasible or fractional solutions that arise from linear relaxations, disjunctive cuts can tighten the formulation by excluding these solutions and enforcing integrality conditions more strongly.
disjunctive. Disjunctive
minor. Derived from graph minor theory, these cuts identify isomorphic substructures in the constraint matrix corresponding to known hard combinatorial subproblems. By recognizing these patterns, the separator generates cuts that exploit the inherent complexity of the substructures.
mixing. Generates cuts by combining multiple weak constraints through coefficient mixing, creating stronger aggregated inequalities. The method systematically blends constraints sharing common variable structures while preserving problem feasibility.
rlt. Eiformulation-Linaazzzze
Technique (RLT) converts polynomial constraints into linear inequalities through variable substitution and constraint multiplication. RLT preserves problem structure while creating convex envelopes for nonlinear terms, enabling strong linear relaxations.
interminor. This separator extends minor cuts by focusing on medium-scale substructures that appear programming (MIP) models. Balances local
as intermediate components in larger pattern matching with global problem structure analysis.
convexproj. Convex projection cuts are generated by projecting an infeasible point onto convex relaxation of the problem and then creating gradient-base based cuts at the projection point. These cuts are designed to improve the separation of fractional solutions in convex nonlinear programs. The method
27

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aims to enhance the solver's ability to make progress by refining the feasible region through gradient information the projected point.
gauge. Geometric cuts based on analyzing the gauge function of the feasible region's convex hull. This separator generates deep cuts orthogonal to the objective gradient by exploiting polyhedral geometry.
impliedbounds. Implied bound cuts are cutting planes that leverage implications between binary and continuous variables to restrict the feasible region of an MILP problem. They enforce tighter constraints by exploiting logical relationships, such as when a binary variable limits continuous variable's ' upper or lower bound.
intobj. Integer objective cuts are cutting planes used in MIP when the objective function is integer- valued. These cuts aim to eliminate fractional solutions from the LP relaxation that are not feasible in the integer solution space. They help tighten the I.P relaxation by leveraging the fact that certain fractional values of the objective function cannot lead to valid integer solutions.
gomory. Gomory cutting planes are derived from the fractional solutions of the LP relaxation of an MIP problem. Once the LP relaxation is solved, any variable with fractional values can be targeted, and Gomory cut is generated to eliminate these fractional solutions, moving the solution closer to integrality. These cuts can be generated iteratively during the branch-and and-bound process to progressively tighten the LP relaxation.
egmip. Chvatal-Gomory cuts generated by forming non-negative integer combinations of the original linear constraints and then rounding the resulting coefficients to produce a valid inequality. These cuts are designed to tighten the LP relaxation by eliminating fractional solutions. The process involves solving a sub-MIP to identify the best combination of constraints, which ensures that the generated cuts are as effective as possible.
strongeg. Strong Chvátal- Gomoory (CG) cutting planes are an extension of the classical CG cuts, which are derived from valid inequalities of the linear relaxation of an integer programming problem. These cuts are used to iteratively tighten the LP relaxation by adding inequalities that exclude fractional solutions. The strong variant refers to CG cuts that particularly effective in reducing the feasible region, leading to a faster convergence to the integer solution.
aggregation. This separator generates cuts by aggregating multiple constraints into single strengthened inequalities. It specializes in flow cover inequalities for network problems, combining are selection variables with flow conservation constraints.
clique. Clique cutting planes are a type of valid inequality derived from the set-packing formulation in integer programming, particularly useful in problems involving binary variables. The inequalities are based on identifying cliques in a conflict graph representation of the problem. A clique is subset of mutually adjacent vertices in graph, representing set of constraints that cannot be simultaneously satisfied. The corresponding clique cutting planes exclude infeasible solutions by enforcing that only one element from each clique can be selected.
zerohalf. Zero-half cuts are specific type of Chvátal-Gomory (CG) cuts in integer programming. These cuts derived using coefficients in (0, 1 instead of integer coefficients. They used to tighten the relaxation of integer programming problems, bringing it closer to the convex hull of feasible integer solutions.
mef. Flow path cuts are valid inequalities used to strengthen the linear relaxation of MIP problems, specifically for problems involving fixed charge networks. They help in modeling flow through sequence of nodes where fixed charges are incurred if any flow occurs along a path. Flow path inequalities operate on constraints related to fixed charge paths, where binary and continuous variables govern the flow through network. These cuts are particularly useful in fixed charge network design and lot-sizing problems. However, the computational consideration is that the structure exploited by these cuts is very specific, meaning they are only applicable to certain problem types.
eccuts. This separator specializes in cuts for edge-concave functions in Mixed- Mixed-Integer Nonlinear Programming(MINNI
generated by constructing supporting hyperplanes at concave function edges. It exploits pnecewsse-linea approximations of nonlinear constraints.
oddeycle. Odd cycle cuts are designed to eliminate infeasible fractional assignments in problems where binary variables represent nodes or edges in a graph. They are particularly effective when
28

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dealing with cycles in graph that contain an odd number of nodes. For example, in a graph-based problem, assigning fractional values all variables an odd cycle is infeasible when the solution must be binary (0 or 1). Odd cycle cuts ensure that such fractional solutions are excluded from the feasible region.
flowcover. Flow cover cuts are a type of cutting plane derived from valid inequalities used to tighten the linear relaxations of MIP problems, particularly for binary single-node flow sets. They are useful in network design and fixed charge problems, where variables can represent flows subject to upper bounds and binary decisions.
emir. MIR cuts are a class of cutting planes derived from mixed-integer sets, particularly when dealing with constraints that include both continuous and integer variables. The main idea behind MIR cuts is to generate valid inequalities by rounding coefficients of mixed-integer constraints to tighten the LP relaxation. They are generated using disjunctive argument, which creates inequalities that separate fractional solutions from the feasible region of the MIP.
rapidlearning. Rapid learning is heuristic technique that temporarily relaxes certain constraints simplifies the problem to solve a more manageable version. By solving this easier problem, the solver gains insights into the structure of the original problem. The rapid learning separator then uses this information to generate useful cuts or constraints that can immediately improve the quality of the LP relaxation in the original problem.
C
Network Details
C.1
The Encoder Network
We provide detailed description of the neural architecture employed in our encoder network. Our design builds upon the framework introduced in L.Sepp [6], incorporating several modifications to enhance performance. The encoder first embeds maps the input features of constraint (C), variable (V), and separator (S) nodes into hidden representations. Subsequently, it performs message passing following the directions of V - C - V. S - V - S, and S - C - S, effectively capturing the interactions among different node types. Then, the S nodes pass through an attention module to emphasize the task of the separator configuration. In contrary to the approach in [45). which outputs score for each cut node, our encoder applies a global additive pooling on each of the C, V, and S hidden embeddings, yielding three aggregated embedding vectors. These vectors are concatenated with two graph-level features, as detailed in Section B.. forming a comprehensive representation. Finally, this combined vector is passed through multilayer perceptron (MLP) to produce unified graph embedding that encapsulates both local and global information pertinent to the problem structure.
D Implementation Details of our algorithm
D.1 Algorithm Pseudocode of DynSep
We employ the Proximal Policy Optimization (PPO) algorithm to train our model. PPO alternates between collecting data through interactions with the environment and optimizing surrogate objective function to update the policy. To ensure stable training, PPO utilizes a clipped surrogate objective that constrains the policy updates. preventing drastic changes that could destabilize learning. Specifically, the policy network is updated to maximize the expected advantage while maintaining the probability ratio between the new and old policies within predefined threshold. Concurrently, the value network is trained to minimize the mean squared error between the predicted value estimates and the actual returns. The training procedure of our method DynSep, including the formulation of PPO objective functions, is detailed in the pseudocode of Algorithm
D.2 Hyperparameters
We train DynSep using the ADAM optimizer [46] within the Py PyTorch framework [77. Consistent with prior studies 33 42,
we split each dataset into the train and test sets with 80% and 20% instances. We train our model on the train set for 100 epochs, and select the best model on the train set to evaluate on the test set. A complete list of hyperparameters for the SCIP solver, the PPO algorithm, and the decoder-only Transformer appears in Table 5
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Algorithm Dynamic Separator Configuration via PPO (DynSep)
1: Denote parameters of the actor's transformer Tx and policy я as 0.
Denote parameters of the critic's transformer TV and value function V as vr Denote parameters of the encoder ① as B.
2: Initialize MII.P instances set X, replay buffer 2, sampling size N,, training epochs Ne. clipping factors learning rates xxx, av and model parameters (0, B).
3: for Ne epochs do
4: Clear the replay buffer D.
5: I/ Data collection
6: for N₂ sampling steps do
7: Randomly sample an instance x form
8:
Run the MILP solver to optimize instance with configuration policy л, collecting N₂ ((N.) Nx from T(N;) separation rounds at Ny nodes.
episodes of data [1000 (0 .r (1))
1==0
i=1
Append collected episodes to D. 10: end for
9:
11:
I/ Model Optimization via PPO
12:
Compute returns RI..... ....... Ry and advantage estimates Å År for each episode in D.
13:
for each minibatch D2 C D do
14:
Compute ratio (0) ED.
(x,))) matta, ()(,)))
15:
Compute actor loss Lactor (0) - EDG Update 8 - 0 ax V Vo (0).
A.).
16:
Compute critic loss Leritic (V. B) - Bo, (v((ss))) - 2.) Update (B. - (V. B) + ag (W Leniti
17:
( (V.
19:
end for
20:
end for
Table Hyperparameters used in DynSep.
Type
Parameter
Value
Number of separators K
22
Frequency of each activated separators fi
10
SCIP Solver
Upper limit of separation rounds 7
5
time limit per instance
300
Training epoch Ne
100
Sampling size N, per epoch Minibatch size [Dpl
16
MIPLIB mixed neos
8
MIPLIB mixed supportcase
6
PPO in Alg.
Other benchmarks
16
Clipping factor & Optimizer
0.2
Adam
learning rates
0.0001
learning rate decay for every kn step
5
learning rate decay rate
0.96
Embedding dimension of attention d
64
Number of attention heads
4
Number of attention layers
4
Transformer
Attention dropout
0.0
Activation of FFN
GeLU
Embedding dimension of FFN dyFN
256
Number of FFN layers
2
Layer number of the linear actor head
Layer number of the linear critic head
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D.3 Hardware Specification
Training and evaluation on the easy and medium datasets were performed on single machine equipped with eight GPUs (NVIDIA GeForce RTX 2080 Tī) and two Intel E5-2667 v4 CPUs (32 logical cores), while experiments on the hard datasets used a single machine equipped with eight GPUs (NVIDIA GeForce GTX 3090 Ti) and two Intel Gold 6246R CPUs (64 logical cores) for hard datasets.
D.4 Baselines
We provide additional implementation details for our baseline methods:
Search(p): This method randomly samples p configurations then applies one with the best performance on the validation The validation set is a subset of the training data, with size equal that of the corresponding test set for each MILP benchmark.
This method deactivates separators with contribution during the evaluation the validation set. Specifically, if a separator's naaiit. Cut Application Rate, DomReds, and Cutoffs (as defined in Section [B.3 are all zero, the separator is deactivated. The validation set used here is partitioned identically to that in the Search(p) method.
Prune:
We configure the SCIP solver parameters in alignment with our default settings, except for the learned separator activation statuses. Specifically, we impose a 300-second time limit, set frequency
L2Sep [6]
10 for all activated separators, and enable both presolve and heuristic mechanisms. The validation set for L2Sep is partitioned identically to that used the Search(p) method. We define the size of the restricted configuration space 15 for easy datasets, 20 for medium datasets, and 25 for hard datasets.
The LLM4Sep baseline utilizes the DeepSeek Chat API to generate separator configurations. The context provided to the language model includes detailed descriptions of each separator, outlined in Section B.5 along with information regarding the problem structures the separators operate on and their computational characteristics. Additionally, the model receives comprehensive description of the MILP problem, encompassing the general formulation and the summary text of MILP objectives and constraints.
LLM4Sep (7]:
E
Additional Results
E.1
Motivation Results on effects of different configurations
Fig. 2(b) in main text shows motivation results for five benchmarks, with alternative names of Dl-Set Covering, D2-Max Independent Set (MIS), D3-MIK, Load Balancing, D5-MIPLIB mixed neos. Here we provide
results for all nine benchmarks.
The left panel of Fig. 5 shows the best perfforfr configuration per instance identified by four randomization strategies applied to separator configurations. It extends the analysis from Fig. 2bb in the main text by presenting results across nine benchmark datasets: D1-Set Covering, D2-Max Independent Set (MIS), D3-Multiple Knapsack, D4-CORLAT, D5-MIK, Anonyyoo,,s, D7-Load Balancing, D8-MIPLIB Mixed Neos, and D--MPLIB Mixed Supportcase. For each instance within these datasets, evaluated 50 random configurations.
The right panel depicts the average performance across instances for each dataset. Our observations reveal substantial performance variance across all four strategies, underscoring the significant impact of the specific separator parameters and dynamic configurations on solver efficiency. Notably, the Priority strategy exhibits comparatively lower variance in performance. This is attributed to the fact that priority adjustments influence only relative ordering separators within a separation round; since the I.P relaxation is not re-solved until the round concludes, such reordering has minimal effect on overall solver performance.
Furthermore, Fig. illlstrraes the impact varying the maximum number of separation rounds per node, denoted as /max- solver performance across nine benchmark datasets. Each plot shows the
31

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Performance Improv. of Best Configs
100%
100%
Average Performace
Solving Time
75%
75%
Intorra
Solving Time
PD Integra
50%
25%
25%
50
10000
0%
D1
02
03
05
D6
08
D9
Datasets
D1
D2
D3
D4
D5 D6 D7
D6
D9
Random Activation
Random Priority
Round
Figure 5: Left: Performance improvement of the best configurations found by different random strategies nine benchmarks. The y-axis represents the relative improvement compared to the default setting. Right: Average performance of configurations sampled by different random strategies on nine benchmarks. The y-axis represents the real performance under two metrics. Specifically, Datasets D1-D5 use solving time (left) as the metric, while D6-D9 use PD integral (right). Each bar represents a specific strategy to get random configurations.
average solving time (red line, left y-axis) and PD integral (blue line, right y-axis) across instances in the dataset. The two metrics exhibit highly consistent trends in each benchmark, indicating strong correlation between solving time and PD integral. The results also show that changes in mmax lead to significant performance variability; however, increasing max does not universally enhance performance, and the optimal value Fmax varies among datasets. Prior work [48] also observes that solver performance is sensitive to the maximum number of cut rounds and learns data-driven stopping policy; however, it does not model per-round separator configuration, whereas we jointly decide when halt and which separators activate each round.
Solving Time
Integral
Maxround (setcover)
Maxround (mis)
Maxround (knapsack)
102
Solving Time
PD Integral
PD Integra
Solving Time
240
24
Solving
5.4
4.8
Fmax
Fmax
Fmax
Maxround (corlat)
Maxround (mik)
Maxround (anonymous)
Solving Time
1.60
Solving Time
Solving Time
100
2/000
1.40
9000
Integre
120
820
PD Integra
24000
25500
integre
400
100
N003
200
22500
2000
21000
Frax
Fyax
Maxround (load balancing)
Maxround (neos)
Maxround (supportcase)
Solving Time
500.136
Solving Time
280
Integra
Solving Time
2.0
2/2
600.120
16003
10000
300.112
256
240
15200
100
L4000
Fmax
Frax
Fmax
Figure 6: Effect of varying maximum round Fmax on solver performance for nine benchmarks. Fach plot shows the average solving time (red line, left y-axis) and PD integral (blue line, right y-axis) across all instances in the dataset.
E.2 Evaluation Results Other Metrics
We provide evaluation results of nine benchmarks for two other metrics of the primal-dual gap (PD gap) and total number of nodes (Nnodes) in Table The PD gap reflects the solution quality achieved by the solver, while Nnodes indicates the size of the B&B search tree- indirect measure of solving
32

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Table 6: Evaluation results nine benchmarks about two other metries of primal-dual gap (PD gap) and total number of nodes (Nnodes), Best performance is in bold. The values report the mean (standard deviation) of time and PD integral metrics.
Easy: Set Covering
Easy: Max Independent Set
Fasy: Multiple Knapsack
Method
PD gap
Nnodes |
PD gap
Nnodes
PD gap |
Nnodes 1
NoCuts
0.0 (0.0)
114.84 (413.86)
0.0 (0.0)
529.31 (1703.82)
0.0 (0.0)
17847.64 .6 (42453.94 944)
Default
0.0 (0.0)
1.11 (1.06)
0.0 (0.0)
1.0 (0.0)
0.0 (0.0)
36,91 (85.56)
Sea
0.0 (0.0)
1.16 (1.59)
(.0))
123.41 (347.52)
0.0
3164.31 (4942.28)
Prune
0.0 (0.0)
207.61 (439.61)
0.0 (0.0)
403.72 (810.19)
0.0 (0.0)
12.19 (34.42)
1)
(0.0)
211.27 (402.0)
(.00)
386,28 (790.7)
0.0 (0.0)
(0495,97 (19385.29)
(R2)
0.0 (0.0)
211.81 (401.94)
(.00)
384.32 (787.77)
0.0 (0.0)
12144.57 (19489.3) .3)
1.I I.MM4Sepasel
0.0 (0.0)
179.04 (412.59)
0.0
32.12 (64.62)
0.0 (0.0)
1883.54 (2884.54)
DynSep (Ours)
0.0 (0.0)
1.0 (0.0)
0.0 (0.0)
1.0 (0.0)
0.0 (0.0)
5.2 (13.59)
Medium: CORLAT
Medium: MIK
Hard: Anonymous
Method
gip 1
Nnodes
PD gop
Noodes
gap
Noodes
NoCuis.
2.67e+18 (6.66-19)
57516.67 (96475.09)
0.02 (0,03)
(1
1.83e419 (3.66=19)
13034.68 (11248 97)
Default
2.73c+19 (1 (1-6c=119)
304.77 (456.1)
0.0 (0.0)
5504,77 (6)75.84)
5,5c+19 (1 (196c-19)
1894.13 (4139.96)
Scarchi50)
4c+18 (1 199966199 19)
35564.78
(88651.06)
0.0
13134,6 (11543,03)
4p+19 (4, 19)
4602. 75 (6696 04)
Prune
4c419)
99579.13
(141189.24)
(0,02)
532615.47
(169443.58)
6.776188 (2.45c+19)
22442.38
(14508.91)
L2Sep(R()
4c+18 (L. 98c-119
84623.58
(117122.52)
0.0 (0.0)
$157.9 (647) ..)
188 (2,19)
15856.25 (14404.64)
L2Scp(R2)
2c+18 (LA1c=119)
$1853.7 (116670.12)
0.0 (0.0)
3616.2 (31 22)
Sc+18 (2. (2.2fc=19)
16151.85 (1 1903 3)
LLM4Sqpaeel
10+18 (1. (1.41c+19)
58808 98. (71956.13)
0.0 (0.0)
5653.0 38)
6o+19 (5.03c-19) 19)
3635,15 (7788. 15)
DynSop (Ours)
(0.0)
4064.16
(0.0)
3076.0 (2666.61)
1,99 (2.96)
15011.2 (8355.63) 63)
Hard: Load Balancing
Hard: MIPLIB mixed neos
Hard: MIPLIB mixed supportcase
Method
PD gap..
Nnodes 1
PD gap J
Nnodes 1
PD gap
Nnodes 1
NoCuts
0.97 (0.12)
1.0 (0.0)
2.5e+19 (4.33et 19)
148834.67 (93798.58)
10.94 (25.26)
2204.17 (0.9)
Default
0.97 (0.12)
1.0 (0.0)
2. (4.33e+19)
12927.5 (16777.1)
2.5e+19 (4.33e+19)
22.25 (54.64)
Scarch(50)
0.09 (0.01)
10.24 (12.67)
2.50+19 (4.33c+19)
35011.0 (44832.71)
0.1 (0.26)
482,12 (729.38)
Prune
0.49 (0.05)
150.51 (65.02)
2,5c+ (4.33c+19)
202703.25 (149382.93)
12.51 (26.94)
6162.54 (13362,48)
L2Sep(R()
0.59 (0.37)
86 36 (55 62)
2.5c+19 (5c+ (9)
66880.0 ( (103828,32) 32)
12:45 (23,27)
1622.38 (3290.69)
1.25ep(R2)
0.59 (0.37)
23)
1. (5e+19)
120425.0 (119640.76 78)
9.29 (17.1)
3414.75 (6492. 98)
LLM4Sepasel
0.12 (0.03)
20.73 (16.85)
2.5e+19 (5c+19)
102753.75 (75635.08)
2.5e+19 (4.6 (4.63e+19)
$7.38 (102.48)
DyaSop (Ours)
0.09 (0.02)
9.99 (11.4)
2.5e+19 (4.33e+19)
119616.25 (105217.02)
(20.31)
69.38 (131.34)
effort, though smaller tree does not necessarily imply faster solving. The results show that DynSep consistently solves all easy and medium instances to optimality, achieving an average PD gap of zero Notably, DynSep is the only configuration method that solves all instances to optimality with zero primal-dual gaps, highlighting the effectiveness of DynSep for fine-grained separator configurations.
E.3 Evaluation on Additional MIPLIB Datasets
We have extended our evaluation beyond MIPLIB mixed neos and mixed supportcase, including two real-world datasets from the Distributional MIPLIB benchmark [49]:
Maritime Inventory Routing Problem (MIRP). MIRP arises in bulk shipping logistics, integrating vessel routing and port inventory decisions under capacity and inventory constraints. Typical instances of MIRP feature average of 15080 binary variables, 19576 continuous variables, and 44430 constraints.
Ceismic-Resilien Pipe Network Planning (SRPN). SRPN involves optimizing municipal water pipe network design ensure resilience under seismic disturbances, targeting service continuity to critical facilities (e.g. hospitals) while minimizing upgrade or restoration costs within budget. Typical instances of SRPN feature on an average of 3016 binary variables, 3016 continuous variables, and 5917 constraints.
The results are summarized in Table 7 For each dataset, we report solving time (Time), primal pual dual gap integral (PD integral), and primal-dual gap (PD gap). All three metrics are lower-is-bett better, where Time and PD integral reflect solver efficiency and convergence speed, and PD gap quantifies how close the solver comes to the optimal. We set the time limit as 600 seconds for each instance. These results show that DynSep delivers marked performance gains on additional real-world datasets (MIRP and SRPN). Compared to the other configuration methods, DynSep significantly improves
33

---

Table 7: Evaluation results MIPLIB MIRP & SRPN Benchmarks, with 600-sccond time limit.
Tand: MIPLIB MIRP (n 34656, m 44430)
Hard: MIPLIB SRPN (n = 6032, 5917)
Method
Time(s)
intogral
PD gap
Timers)
PD integral 1
PD gap
NoCuts
486.68 (198.71)
54735.55 (19014.81)
6.67c+18 (1.92c+19)
280.7 (277.17)
11020.69 (13133.86)
0.28 (0:42)
Default
580.98 (61 65)
52362.46 (13476, 17)
5.38c+ (4.97c-19)
332.04 (271 0)
11687.0 01 (11993.95)
0.21 (0.31)
Search(20)
492.85 (190.1)
33028,81
(16299.41)
6,30c+18 (2.11c+19)
384.02 (273.02)
14571.56 (12206.64)
0.26 (0.28)
Prune
501.49 (174.33)
35542.82 (18572. 11)
8.33c+18 (2.19c+19)
296.08 (277.77)
9211.76 (10664.7)
0.19 (0.26)
LLM4Sepasel
511.97 (161.003
34573.52 (18681.26)
5e+18 (2.24c+19)
300.66 (284.75)
8421.65 (10209.65)
0.14 (0:22)
DynSop (Ours)
482.13 (205.78)
30838,39 (17377.22)
1.38 (1.41)
294.77 (274.22)
7581.16 (8814.39)
0.1 (0.17)
Table 8: Comparison between default setting and our method (DynSep) on all 235 MIPLIB 2017 instances.
Hard: MIPLIB 2017
Method
Time(s)
PD integral J
Default
258.77 (93.22)
17153.69 (12674.03)
DynSep (Ours) 238.88 (107.33)
15092.16 (12719.67)
both convergence speed (as demonstrated by reduced PD integral) and solution quality (evidenced by lower PD gap).
Furthermore, have tested method the full set of 240 MIPLIB 2017 benchmark instances. The results in Table 88 show that DynSep delivers notable improvements in solving efficiency in the challenging MIPLIB 2017 dataset.
Specifically, we set the time limit as 300 seconds and excluded five instances whose presolving time exceeded 300 seconds: neox-33402454-bohhe
neos-471244-wwidder mzzv42z, meos-5052203-ccyvv proteindesign/22 2/222 /2p9, and proteindesign /22rrx/ Ip8, which is common removing criterion for MIPLIB2017 benchmark 18 42]. The remaining 235 instances were split into 70% training set and a 30% test set. Table reports the overall average performance of our method across all 235 instances. These experiments confirm that our approach delivers notable improvements in solving efficiency, even when evaluated on the more challenging benchmark set.
E.4 Extended DynSep to Broader Solver Hyperparameters
Our proposed DynSep framework is inherently extensible to broader set of solver hyperparameters beyond separator activation and timing. This is achieved by adapting the policy network: we model the outputs of additional parameters logistic-normal distribution, followed by optional discretization to support both integer and continuous parameters. This enables flexible, differentiable control of arbitrary solver parameters.
To substantiate the above claim, we conducted additional experiments on three critical hyperparameter groups as follows, while retaining the original tuning mechanism for separator activation status (+1,0, 0, and round termination (m,)
para group 1: Cut Depth Dewpth/Agreeeeeee parameters
Aggressiveness (sepastore/age in SCIP). Controlled by solver cutttinegngtttggimiiiii and separating/poot freq.
para group 2: Cut selection thresholds (e.g., efficacy vs. orthogonality). Controlled by wwweeettiioobbrr//miiieei
and para group 3: Separation frequency per node. Controlled by solver parameters
solver parameters separating/mineffiicc
separating//cutaglliim and separating/poolffi
We provide the experimental results in Table We evaluated our method on three benchmark datasets: Multiple Knapsack, Corlat, and MIPLIB mixed supportcase, measuring solving time (Time), primal-dual gap integral (PD integral), and primal-dual gap (PD gap). All three metrics are lower-is-hetter, where Time and PD integral reflect solver efficiency and convergence speed, and PD gap quantifies how close the solver comes to the optimal. We compared our original DynSep method with its extended versions incorporating three additional parameter groups. Results show
34

---

Table 9: Experiments three extended hyperparameter groups configured by DynSep.
Easy:
Mutiple Keepeack 720,
Mediam
430)
mused suppectiase
Method
Tinelo
PD Interral
Timero
Timelst
PD
PD
DynSep (Curr)
(:.200:24)
9.71 15.29)
00:00.00
22.86 158.5)
2233.42 (3564 431
00:00:00
332.50
9212.24 (964))
7.86-22 31)
Para Grapp
6.6510.39)
9.11
0.000.00
4ce 8
(0:25)
Group
7,91
1922.77 (3826.2)
0.0 (0002)
09 (125.30)
8362.53 (6834.04)
Table 10: Execution Time for each decision step of DynSep to configure separators
Set Covering
MIS
Knapsack
CORLAT
MIK
Anonymous
Load Balancing
Neos
Supporicase
Avg. Latency (s)
0.33
0.22
0.09
0,05
0.31
0.41
3.04
0.11
0.39
Max. Latency (8)
1.09
1.01
1.07
1.94
2.02
6.85
that across all datasets, DynSep and its extended variants consistently outperform the default solver configuration reported in the main paper. Furthermore, the differences among the parameter groups are relatively small, yet certain combinations (e.g.. Para Group on Knapsack and Para Group on mixed supportcase) yield further performance improvements in both Time and PD integral. These results validate that DynSep can flexibly extend control broader set of solver hyperparameters. Furthermore, carefully chosen parameter combinations can yield additional gains in solving speed and convergence.
E,5
Overhead Evaluation
E.5.1
Latency of Policy Inference
We have provided per-decision latency (the latency of policy inference per decision step) and the total inference time through the solving process as follows.
Per-decision latency. Table T0 reports the average ("Avg. latency") and worst-case (" ("Max. latency") time taken for a single policy call across the entire branch-and-cut process. These values reflect the inference latency introduced by DynSep policy for each configuration decision.
Our results reveal that per-decision latency increases instance size grows. On small to medium-sizee datasets, the worst-case latency remains around second per policy call. For the larger and more complex problem instances (with tens of thousands of variables and constraints), the worst-case latency ranges from to 7 seconds per decision. In contrast, the average inference latency stays below 0.5 seconds across all datasets, with the exception of the largest load balancing instance (approximately K1K variables, 64K constraints), where the average latency rises to 3 seconds.
Table I1 provides the total inference time ("Infer. Time") over difffrrent datasets and solver time limits, along with the inference overhead rate ("Overhead Rate"), which represents the percentage of policy inference for the total solving time ("Sol. Time"). Table IT shows that DynSep incurs modest configuration overhead, contributing a negligible fraction of the total solving time even on large-scale instances. While the configuration time tends to increase with problem size-e.g. from under one second on small benchmarks to several tens of seconds on the largest ones it remains practically affordable relative to the performance gains achieved. Nonetheless, there is potential to further optimize this step, and future work may explore more lightweight models to reduce configuration latency without compromising effectiveness.
Notably, our configuration policy is encapsulated within a custom SCIP separator. Thus, we report the total inference time via SCIP's built-in SCIPsepaGe Time () function. In contrast, the per-decision latency values are logged using wall-clock timing at each policy call, including overhead from time recording and logging. Consequently, the sum of the per-decision latencies naturally exceeds the total inference time, since the latter excludes the additional overhead introduced by frequent timing operations.
E.5.2 Memory Overhead
We analyze the memory inference overhead as follows.
Per-decision memory overhead: Table 12 below show the peak GPU memory usage per policy call.
35

---

Table 11: Inference Time for DynSep to configure separators during solving process. Three blocks: nine datasets in our manuscript second time limit), four hard datasets (3600-second time limit), and MIPLIB MIRP & SRPN (600-second time limit).
Coverin
MIS
Krujoakk
CORLAT
MIK
Ansuymmus
Load Baluncing
Suppericas
Infer. 105 (4.34)
(0.23)
(.2(0.51)
(20.58)
(0.33)
(27.59)
(19.65) 132.50 (130.32)
Sol. Time
(7)
(0.20)
0.52
(3892)
(9.999444))
241.89 (100.75)
235.19 (112.20)
Overhead (%)
09.54
77.36
80.77
72.13
6.09
Anonymous
Load Balancing
MIPLIB mixed neos
MIPLIB mixed supporicase
Infor. Time (4)
21.32 (16.64)
16.01 (1.57) 3600.04 (0.07)
86,96 (114.55) 2724.85 (1515.82)
12:31 (9.9) 567.82 (1154486)
Sol. Time (8)
2397.95 (1551.2)
Overhcad Rate (1k)
0.89
0.44
2.17
MIPLIB MIRP
MIPLIB SRPN
Infer. Time (s)
32.77 (29.27)
5.06 (2.28)
Sol. Time (s)
482. (205.78)
294.77 (274.22)
Overhead Rate (%)
6.80
1.72
Table 12: Per-decision memory overhead.
Set Covering
MIS
Knapsack
CORLAT
MIK
Anony
Load Balancing
Ncos
Supportcase
Avg. CPU (MM)(MM)
2902.22
28/4.220
2807 .59
2786.58
2775.91
3153,15
3811
2824.
2870.90
Max. CPU mem. (MB)
3081.70
2877.39
2889.04
2592.15
2877.77
3930.07
4852.27
3100.90
3688. 11
Avg. GPU mem. (MB)
2.09
2,09
2.10
2,09
2.10
2.09
2,10
2,09
2,09
Max. GPU
2.10
2.11
2.11
peak memory overhead for overall inference: The memory footprint required to store the encoder and policy model weights is 2.08 MB for each dataset. We track the peak GPU memories during the reference via the tool torch. cuda. wtt.maa_mmeee__e_ www.mmmoooooloooeeeeeeee which are list in Table [3]
E.6
Ablation Study
E.6.1 Module Ablation Analysis on Other Six Benchmarks
We evaluate DynSep and its ablated variants on other six benchmark datasets using solve time and the primal-dual (PD) gap integral as performance metrics. Table 14 summarizes the results. Specifically, the ablation study shows that while certain DynSep variants (c.g., w/o DynG&TF in Set Covering, w/o TF in MIS, w/o MaxR in Anonymous and Load Balancing) can slightly beat the full model on individual metries for particular datasets, the full DynSep method consistently delivers robust, near-best performance overall. Overall, each component's s removal yields trade-offs, but the full DynSep model demonstrates consistently balanced performance across tasks.
E.6.2 Encoder Architecture Ablation
Table 15 shows that replacing the encoder's GCN with a custom bipartite graph transformer (GT); melti-head TransformerConv for edge-aware message passing, followed by residual-connected LayerNorm and a two-layer feed-forward block. GT-encoder shows no consistent improvement over DynSep, which may be due to increased inference overhead or the need for finer stability/tuning to realize gains from the transformer-style aggregation.
E.6.3 Hyperparameter Sensitivity Analysis
As shown in Table 16 we conducted robustness ablations over separator frequency (1, 5, 10, 20) and maximum separation rounds (3, 5, 10, 20) on three benchmarks (MIK, Corlat, and MIPLIB mixed Neos). Overall, across almost all tested settings, DynSep outperforms the default configuration, showing that the approach is reasonably stable to these hyperparameters. Below is our key observations.
Frequency. Moderate frequency (e.g.. 5&10) gives the better trade-off. That is, small frequency causes separators to fire excessively, incurring high cut-generation overhead, whereas large frequency reduces opportunities for timely dual-bound tightening.
36

---

Table 13: Peak memory overhead for overall inference.
Set Covering
MIS
Knapsack
CORLAT
MIK
Anonymous
Load Balancing
Noos
Supportense
Peak GPU mem. (MB)
35.56
$7.85
23.11
15.68
37.41
157.17
257.60
28.85
122:28
Table 14: Ablation results on other six benchmarks
Easy; Set Covering 1000,
Basy: Independent Set
500, 1953)
Mediumi MIK 413, 346)
Methed
Time(s)
Engeo::
PD integral
Time(
(time,
PD integrall
Time(s) 1
Improw.
PD integral
NoCuts
7.45 (5,77)
NA
(55.59)
15.32
(56.99)
190.28 (113.97)
NA
587.85 76)
Defomit
(1.99)
29.66
95. 36 (36.85)
30.4 08 02)
-98.43
289.51 (103.81)
16.65 (18.06)
91.25
82.88 (56.24)
wlo MuxR
1.52 (0.72)
82:28
31.35 (11,32)
0.57 (0
10:46 (2.83)
12:46 (8.81)
93.45
128.9] 89)
w/o TP
1.48 (0.35)
80.13
32.86 (6.25)
0.44 (0.09)
97.13
(1.77)
14,04 (12.99)
92.02
106.14 (65.99)
wia Dynf
1.25 (0.68)
83.22
29 (10. 34)
0.56 (0 (0.22)
96.34
10.19 (2.68)
11:58 (9:44)
93.76
116. 16 c40 (44)
w/o DyaG&TF
1.23 (0.89)
83.49
29.33(110:2)
0.55 (0.15)
96.11
9.97 (2.28)
(10.3))
93:47
127.63 (66.03)
DyuSep (Ours)
1.53 (0.27)
79.73
53.88 (9.34)
0.53 (0.20)
96,54
9.66 (2.48)
10.99 (9:44)
94.22
134.15 (44.21)
Hard Asseecaces 37851, at 43003)
Hard Load Belansing 61 51000, 64394)
Had: MIPLIB mixed suppoctiase (n- 19765, 19910)
Mathol
Timzis)
integnd
Inpers:
Ist..
Timd
P1 iniggral
Impeos.
Ist.
Time
PD integga 1
Ingrr::
(4)
250 (75.71
21117
19254.01
11 (0.02)
15t63
26 (120 25)
12:59 (10005.4))
NA
Detiit:
248,92 (4 (9)
27059 54 (450 80
26.19
3001 14 (0.02)
15167. 19: (936 38)
-0.62
244.75 (105.0
2156 (M (0434.42)
-56.7
who Masit
14482.24
(9648-56)
$281,9 (454. 961
43)
10158.06 (9M8) 77)
58)
(9952-00)
Dysks
256,66 (77.34)
16903.7
(884).95)
19:55
(0.02)
15000.22 (94)
145,
(121 44)
11882.57
9.31
with DenG&TT
346-60 (01.0.1)
15914.52 52 (8:59.911
10-43
300 (6.04)
5923 ($99.21)
14.00
131.72 7.1.72211 (1.3.92)
11369
(12085 (000
12:37
Dysthp (Dund)
241 89: (100.75)
15656. 7 (8996) 14)
25:85
300106 (5.08)
1720:26 (899.77)
75.35
132, (1 50.55)
9212.34 (9848.56)
25.02
Maximun Separation Round. Setting this value too low produces weak cuts and degrades performance, while setting it excessively high increases the computational cost of cut generation. Although this parameter shows some dataset sensitivity, MaxRound=5 empirically near-optimal in our tests.
E.7 Generalization Study
We investigate complementary generalization performance of our method under two more settings: (I) Fross-Domain Generalization Test: training on one benchmark and testing on dataset from a different domain; (2) General-to-Speciff Generalization Test: training general model on mixed-catee mixed-category dataset (e.g. MIPLIB) and then evaluating on specific class dataset.
E.7.1 Cross-Domain Generalization Test
To evaluate the across-domain generalization, we train our policy on one problem family and apply the learned policy to unseen problem families. Specifically, we train four separate DynSep policies (on Setcover, Knapsack, MIK, and Supportcase) and evaluate each of them across all nine benchmark families used in our manuscript. The table below lists the results, where we report solving time for easy and medium datasets, while additionally report primal-dual integral (PD integral) for hard datasets.
As shown in Table 17 policies trained on one problem type yield improvements over SCIP's default in most unseen benchmarks, indicating effective transfer of our learned configuration strategy across NP-hard families.
E.7.2 General-to-Specifte Generalization Test
We select 168 diverse instances from the MIPLIB 2017 benchmark 14 as our training set and learn a separator configuration policy on these instances. Notably, because the MIPLIB 2017 instances cover a wide variety of problem types and mixed- scenario structures, this learned policy could serve as general configuration model. We then evaluate it it-without any additional tuning-op on four unseen, domain-specific datasets: Corlat, Load Balancing (LB), Maritime Inventory Routing Problem (MIRP), and Seissic--Resilies Pipe Network Planning (SRPN). We have extended our evaluation beyond MIPLIB mixed neos and mixed supportcase, including two real-world datasets from the Distributional MIPLIB benchmark [49]:
37

---

Table 15: Comparative results of different encoder architectues for DynSep four datasets.
Easy: Multiple Knapsack (n = 720, m 72)
Medium: Corlat (n = 466, m 486)
Improv. 1
Improv. (time, %)
Method
Time(s)
PD integral U
Time(s)
PD integral J
(time, %)
GT-encoder
0.71 (0.39)
94.87
11.02 (5.41)
46.26 (71.4))
29.04
4563.26 (7125.57)
DynSep (Ours)
0.52 (0.24)
96.24
9.71 (5.39)
22.96 (38.93)
69.25
2233.42 (3868.43)
Hard: MIPLIB mixed neos (n 6958, 5660))
Hard: MIPLIB mixed supporicase (ir 19766, 19910)
Imprrv. 1
Imprrow 1
Method
Time(s)
PD integral
(PD Int., %)
Time(s)
PD integral 1
(PD Int., %)
GT-encoder
243.52 (97.82)
12134.57 (12142.03)
16.99
150.38 123.13)
9157.61 (9623.27)
29.34 28.92
DynSep (Ours)
238.19 (112.26)
8511.58 (12413.9)
41.78
132.50 (130.32)
9212.24 (9840.56)
Table 16: Sensitivity Analysis of hyperparameters Frequency and Maximum Separation Round on three datasets.
Medium: MIK (n 413, 346)
Medium: Corlad (1 466, 456)
Hand: MIPLIB missed noce 6955,
5660)
Mathod
Timejj) 1
Ingros. (linc, %)
U13 integral 1
Timelo) 1
Ingar Oine: 9))
PD integral 1
Timets) 1
PD integral
Inpros. (PD Ist,, 9)
NoCuns
100.28 13.07)
(839
(12.229)
2687 68 48)
225 (43.29)
14618.53 (1221463)
Desialt
16.55 (8066)
(4)
111.55 (132.19)
10573.14
282.95 (29.49)
18500.5 (9386.
Frogr1
4582. 90
252,
$6.55
243
Praq-10 (Duns)
10.99
134.15 (44.2)
22.96 (38.93)
2235.42
(3868.45)
235.19 (112.26)
85111.5 (1241399)
41.78
Preq=20
12,83
95.26
213.35 (1244.09
49.14 (72.87
4638.99 (7252.41)
201
87299 (12229, (6)
29.87
Mediumm MIK = 413, = 946)
Mathum: Corbar to 466, 436)
Hanl: MIP Jli miszd movs 6958 5660)
Method
Timela 1
Empros. 1
(iiee 14)
PD integral
Tine6s) 1
Engno. 1
cime, (4)
PD integ:al
Timelsa 1
PD integral 1
Ingos: Im (h)
18:25
367
NA
Defuit
10573
M8
18500
30.50
MacRounde3
13.63 (12,04)
92.94
150.19 (46.7777 77)
167.26 (124.27)
35.66
SING 26
(1.4011 1855
239 22 (105, 28)
$581.5 (12576.07)
41.30
Machband-3 (Dan))
6L99 (9.44)
94.22
134. (44.21)
22.96 (58.93)
69.25
2233.22 (M66444)
238.19 (112..5))
$511. 5% (12411
41.78
Mackound=10
(12602)
54.75
[82.71 (12)
40.25 (00993)
45.82
3717
64 (67644777)
249 61
8866 91 (12224 34)
39:48
Maclounn-20
(11.5)
91.49
(71.61)
36.71 (55.3) (55.75)
30.83
(5562.511
(114466.36
As shown in Table 18 the general configuration model consistently outperforms the solver's default settings in both solve time and convergence behavior, demonstrating good generalizability of our method.
E.8 Visualization of Separator Configurations on Nine Benchmarks
We provide visualization of separator configurations on nine benchmarks in Figs. 15 Figs. 7 and
show that our learned policy uniformly uniformly reduces the maximum number of separation rounds Fmax 3 for easy benchmarks, Set Covering and MIS, demonstrating that learned decision on maximum rounds effectively prunes unnecessary cutting rounds on simple problems. The heatmap reveals that the separator configuration is not static but varies dynamically across separation rounds (shown along the y-axis), suggesting that the model is timing the application of various separators to coincide with the stage of cut generation. Furthermore, the fact that learned activation values are not restricted to (-1. 0, 1} but take intermediate real values indicates the policy differentiates between individual instances (and even between nodes) when selecting separators. In other words, it has learned nuanced, instance- wise (and node-wise) cutting strategy rather than a mee-size-fite rule.
38

---

Table 17: Cross- domain generalization on nine benchmarks. Policies are trained on one problem family and evaluated on unseen families.
Easy: Set Covering (e = 10m0 5009
Easy: Max Independent Set 500, 1953)
Easy:
Muttipic Knepsock
721l, 72)
Method
Time(s)
PD integral
Timeis) 1
PD integral
Timels)
PD intograi
Defuilt
(1.79
95.56 (16, 86)
30,4 (8 (2)
289.51 (105.81)
2,01 (1.82)
(8.5(10.49)
Thuin on Setcower
NA
NA 40.96 (9.93)
1.0 (1.38)
13:52 99)
0.88 (0.42)
(6.13
Train Knapaack
2,02 (0,62)
0.76 (0.29)
12:37 (3.53)
NA
NAA
Trai MIK
7.74 (4.33)
80.73 (9.35)
29:51 (19.2)
(1,00
(5.52)
supporicse
(0.59)
(0.3)
11.72 (3,63)
(5.97)
Medium: Corlat (x 466, w 486)
Medium: MIK (n 413, An 346)
Hard: Anonymous (n - 37881, mI 400015)
Mathod
Time(s) J
PT integral
Time(s) 1
PD integral
Time(s) 1
PD integral
Default
111.55 (132.19)
10573.14 (13070.46)
16.65 (18.06)
82.80 (56.24)
298. 92 (4.09)
27069 58 (1892.8)
Train on Setpover
43.79 (76.54)
4042.34
(7652.31)
24.8 (21.41)
1.39. I (45 .5)
250.46 (87. .1)
1970 65 (9639)56)
Train Knapsack
27.56 (58.48)
2485.67 (5777.55)
12.48 (10.29)
139.06 (44.39)
253.74 (80.42)
19183.3 (8941.1)
MIK
(89.18)
2163.38 (4884 18)
20715.22 25 (67)
Train on supportcase
20.94 (50.99)
2016.37 (5098.97)
163.6 (98.18)
785.46 (577.89) .899
256.33 (79.04)
17922.89 (8384.44)
Hert Toos naloncing in 61000, w: (4100)
Bond: MIN IN mooes (o 69NW at 1100))
Hest MIP JIN misd sepporoose ON 1976, ve: 10990)
Mothal
Timela 1
PD Innggaa ]
Timetoy
PD inaggral 1
Timeto)
integral
Delait
(0.14
1S162 19 (916 58)
362 (29-4))
18500.5 (8 19196.155
344 (5 81
21581 (0055444)
Trois en Senowwer
M0005 (8:05)
4811.86 (584.3))
158 (12
13112 77 (17MAA 76)
141-31 (125,51)
13171 62 (11599
Krsquack
300.4: (0.85)
4583.45 (554.52)
(75
13343.38 (12584/84)
156.9 (116.10)
15218.09 (11291.29)
Tran aepurizos
(0.09(0.M)
9023.04 6% (065) 17)
246.91 (91.96)
196.99 71 (124-10 81)
NA
NA
Table 18: Generalization performance of our DynSep model trained on MIPLIB 2017, evaluated on four unseen MILP scenarios. (300-second time limit for Corlat & LB; 600-second for MIRP & SRPN)
Medium: Corlat (n 466, m 486)
Hard: Load Balancing (n1 = 61000. m 64304)
Method
Time(s) 1
PD integral
Time(s) 1
integral L
Default 2.01 (1.82)
5.24 (1.79) 18.6 (10.49)
95.56 (36.86)
30.4 (8.02)
289.51 (103.81)
DynSep (ours) trained MIPLIB 2017
46.05
4289_33
300.08
4792.71
Hard: MIPLIB MIRP
34656,
44430)
Hard: MIPLIB SRPN
(n 6032, 5917)
Method
Time(s) 1
PD integral
Timo(s) 1
PD integral L
Detault
580.98 (61.65)
$2362.46 (13476.17)
332.04 (271.0)
11687.01 (1 (11993.95)
DynSep (ours) trained on MIPLIB 2017
487.59
33193,25
7582.60
Sepa, Config at Each Sepa. Round (setcover)
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Figure 7: Separator configs at each separation round of Set Covering benchmark.
Sepa. Config at Each Sepa. Round (mis)
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Figure& 8: Separator configs at each separation round for Maximum Independent Set benchmark.
39

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Sepa. Canfig at Each Sepa. Round (knapsack)
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dijundiee
comwergn
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Figure 9: Separator configs at each separation round for Multiple Knapsack benchmark.
Sepa. Config at Each Sepa. Round (corlat)
Seperation round
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Figure 10: Separator configs at each separation round for CORLAT benchmark.
Sepa. Config
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geuee
Jerorry
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appegattoon
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oaloyiie
Figure 11: Separator configs at each separation round for MIK benchmark.
Sepa. Config at Each Sepa. Round (anonymous)
Separation round
Actvvatio Status
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Figure 12: Separator configs at each separation round for Anonymous benchmark.
40

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Sepa. Config at Each Sepa. Round (load ba balancing)
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Figure 13: Separator configs each separation round for Load Balancing benchmark.
Sepa. Config at Each Sepa. Round (miplib mixed neos)
Separatioon round
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Figure 14: Separator configs at each separation round for MIPLIB mixed neos benchmark.
Sepa. Config
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Figure 15: Separator configs at each separation round for MIPLIB mixed supportcase bench- mark.
41


Reviews:
[{"review_id": "n1ctkyrrL2", "reviewer": "Reviewer_ZkPt", "summary": "This work proposes DynSep, a reinforcement learning–based method to dynamically control how cutting planes are added during the Branch-and-Cut process in solving MILP optimization. The authors claim that instead of relying on fixed or manually tuned separator settings, DynSep learns to decide both which separators to activate in each separation round and when to stop adding more cuts at each node while adapting these choices as the problem evolves across different rounds and nodes. To keep it efficient, DynSep only looks at the incremental subgraphs added in each round, then using GCN and a decoder-only Transformer to track the context from earlier rounds and predict the next separator configuration. The authors claim this design helps the solver avoid unnecessary separation rounds, capture useful separator interaction patterns automatically, and improve both speed and solution quality. In experiments, DynSep reduced average solve times by up to 64% and improved solution quality on harder benchmarks, while also generalizing well to larger, unseen MILP instances.", "questions": "- It is unclear how the GCN encoder is trained, is it included in the transformer training? \n- How might DynSep be adapted or extended to handle nonlinear or mixed-integer nonlinear programming (MINLP) problems?\n- How does the blocked positional encoding impact performance?\n- Could alternative architectures such as graph transformers improve the modeling of separator dynamics?\n- Is the training based on the same types of MILPs in the test benchmarks? How well does it generalize to unseen types MILPs?", "limitations": "- The method is limited to MILP problems and does not cover nonlinear formulations that frequently occur in real-world applications.\n- The method maybe limited to the types of MILPs seen during training, and generaliability to unseen types of MILPs is unclear.", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 3, "significance": 4, "originality": 4, "strengths_and_weaknesses": "## Strengths \n\n- The main strength of this work is that DynSep is an adaptive method that learns not just which separators to use but also when to stop adding cuts, adjusting dynamically as the Branch-and-Cut process evolves. \n- Its design is efficient, since it processes only new cuts each round and uses a Transformer decoder to capture temporal context. \n- The proposed DynSep has demonstrated consistent and significant improvements in runtime and/or primal-dual gaps on various benchmarks, and datasets. The ablation study also helps with interpretability. \n- Built on the open-source SCIP solver, DynSep seems to be also practical and reproducible.\n\n## Weaknesses\n- There is a lack of clarity in the formulation of the solutions. For example, it is unclear the difference between cutting planes and separators until the very end of experiments. As a key concept in the solution and graph model, the separator and how it generates cutting planes are not explained, and it is unclear if the set of separator nodes remain the same for each triplet graph. It is also unclear how separator nodes are represented in the triplet graph, do they have node features or simply represented by node identity. \n- The transformer generates an action based on the history of token blocks. But the tokenization or encoding of the vectorized action is unclear in the paper. Is the action a sequence of tokens or a single token? If an action is a token, is the token space the same as the action space? \n- Within each block, node embeddings have identical positional encoding. How about graph positional encoding such as spectral PE from graph transformer literature? If the authors deliberately choose not to use graph PE, some explanation would be good. \n- Many notations are used before their first introduction, such as PD integral, MDP, Node and Round-wise dynamic configurations. \n- It is unclear about the generalizability of the DynSep due to the lack of training information. Is it trained on the same types of NP-hard problems in the benchmarks? If so, how well does it generalize to other unseen benchmarks? This could be a major limitation and need more clarification.", "was_revised": true, "final_justification": "The authors have addressed my concerns on clarity of presentation and generalizability of the proposed method across problem sets. Therefore I raise my original rating of 4 to 5.", "rebuttal": "We thank the reviewer for the positive and insightful comments. We respond to each comment as follows and sincerely hope that our rebuttal will properly address your concerns. If so, we would deeply appreciate it if you could raise your score. If not, please let us know your further concerns, and we will continue actively responding to your comments and improving our submission.\n\n**Q1. Lack of clarity in the formulation of the solution.**\n\n**A1.** \n\nWe apologize for the confusion. Below, we provide the detailed specifications as suggested, and we will include them in the revised manuscript. \n\n1. **Concept of separators.** Separators are core components of a mixed‑integer programming (MIP) solver. Each separator implements a dedicated cutting‑plane algorithm to generate a family of valid inequalities (i.e., cuts) to tighten the LP relaxation. For example, the *Gomory cut separator* derives Gomory cuts from the current simplex tableau to remove fractional solutions.\n2. **Difference between cutting planes and separators.** Cutting planes are valid inequalities that tighten the LP relaxation, which exclude fractional solutions while preserving all feasible integer solutions. Separators are built-in algorithms that generate these cutting planes.\n3. **Separator nodes.** In our triplet graph, each separator is represented as a dedicated node. **Separator node features** comprise a one-hot encoding of its type, plus the separator's performance statistics (e.g., execution time, number of applied cuts, and dual-bound improvement) to capture historical efficacy. Full definitions of these features appear in Table 4 of the appendix. **Notably, the set of separator nodes is fixed** (corresponding to 22 SCIP’s built‑in separators) and remains the same for every instance, while their feature values are updated dynamically at each decision step.\n\n**Q2. How are actions tokenized/encoded for the transformer?**\n\n**A2.**\n\nEach action $a_t = [m_t, \\eta_{t,1},\\dots, \\eta_{t,K}]\\in\\mathbb{N}^{K+1}$ could be viewed as a block of tokens, consisting of one token for the maximum round threshold $m_t$ and $K$ tokens $\\eta_{t,1},\\dots, \\eta_{t,K}$ for $K$ separators' activation status. Below, we describe the tokenized formulation of the action.\n\n+ **Maximum‑round token** $m_t$: drawn from an $r_{\\max}$‑dimensional categorical distribution (its token space is $ \\\\{1,\\dots,r_{\\max} \\\\} $).\n+ **Separator‑activation tokens** $\\eta_{t,i}$: for each separator, one token drawn from a 3‑dimensional categorical distribution over $ \\\\{-1,0,1\\\\} $ indicating three distinct activation statuses.\n\n**Notably, we clarify that we do not feed the sampled action $a_t$ back into the transformer.** Instead, we execute the action $a_t$ in the solver environment and obtain the next incremental state $\\Delta s_{t+1}$ (i.e., the incremental graph). We input $\\Delta s_{t+1}$ as the subsequent token block into the transformer. This yields an autoregressive loop for both the training and reference phases--each next-step prediction $a_t$ conditions on the entire history of past incremental states $\\\\{\\Delta s_{1:t}\\\\}$ (equal to $s_t$). Such procedure matches the modeling of standard online RL policy $a_t=\\pi(s_t)$, where the policy $\\pi$ depends only on the current state $s_t$, not on prior actions.\n\n\n\n**Q3. Why not use graph-based positional encodings (PEs) (e.g., spectral PE)?**\n\n**A3.**\n\nWe acknowledge that, since all separator nodes are fully connected to variable and constraint nodes, they share the same graph topology, thus receiving identical spectral PE. **However, we deliberately opt for blocked PE for two main reasons**:\n\n1. **Computational Efficiency:**\n   Computing spectral or other graph‑based PEs requires eigendecomposition or additional operations on the graph Laplacian (or related matrices), which **scales poorly on large graphs**. In contrast, our blocked PE leverages the prior knowledge that separator nodes have no inherent sequence order and is aligned with minimal overhead.\n2. **Dynamic Temporal Modeling:**\n   Graph‑based PEs encode static topological similarity and **cannot capture the evolving, time‑series nature** of the dynamic separator configuration. In contrast, our blocked PE preserves the chronological order of each incremental subgraph while treating separator nodes as an unordered set. This ensures the model focuses on the temporal progression of separator activations rather than a fixed structural pattern.\n\n\n**Q4. Several notations appear before their first introduction.** \n\n**A4.**\n\nWe appreciate this feedback. We’ll move those definitions earlier for clarity.\n\n\n**Q5. The generalization of DynSep.**\n\n**A5.** \n\nThank you for raising this point. In our evaluation,  DynSep policies are trained and tested within the same problem class.\n\n+ **To evaluate the across-domain generalization**, we train our policy on one problem family and apply the learned policy to **unseen problem families**. Specifically, we train 3 separate DynSep policies (on Knapsack, MIK, and Supportcase) and evaluate each of them across all nine benchmark families used in our manuscript. The table below lists the results, where we report solving time for easy and medium datasets, while reporting PD integral for hard datasets.\n\n  As shown in the table, policies trained on one problem type **yield improvements over SCIP’s default in most unseen benchmarks**, indicating effective transfer of our learned configuration strategy across NP-hard families.\n\n||SetCover|MIS|Knapsack|CORLAT|MIK| Anonymous   |LB|Neos|Supportcase|\n|-|-|-|-|-|-|-|-|-|-|\n||Time (s)|Time (s)|Time (s)|Time (s)|Time (s)|PD integral|PD integral|PD integral|PD integral|\n|Default|5.24|30.4|2.01|111.55|16.65|27069|15187.19|18500.5|21561.09|\n|Train on Knapsack|**2.02**|0.76|/|27.56| **12.48** |19183.3|**4583.45**|**13343.48**|13216.93|\n|Train on MIK|7.74|5.21|1.2|34.54|/|20715.25|5559.96|13366.13|**11583.1**|\n|Train on supportcase|2.21|**0.67**|**0.78**|**20.94**|163.6|**17922.89**|9421|13639.73|/|\n\n+ To further demonstrate generalization, **we also evaluated on real-world datasets**. Although they come from the same application domain, geographic, temporal, and related variations induce significant distribution shift, thus cross-dataset performance could serve as **a proxy for transfer to unseen problems**.\n\nAs shown in the table below, our method (**DynSep**) delivers **marked performance gains** on two real‑world datasets (MIRP and SRPN) from the Distributional MIPLIB benchmark [1], providing evidence of its ability to generalize to unseen, distribution-shifted real-world instances.\n\n|Instance|mirp|mirp|mirp|srpn|srpn|srpn|\n|-|-|-|-|-|-|-|\n|Metric|Time(s)|PD integral|PD gap|Time(s)|PD integral|PD gap|\n|Default|580.98|52362.46|5.38e+19|332.04|11687.01|0.21|\n|DynSep(Ours)|**482.13**|**30838.39**|**1.38**|**294.77**|**7581.16**|**0.1**|\n\n\n**Q6. How the GCN encoder is trained**?\n\n**A6.**\n\nWe use separate decoder-only Transformers for actor and critic. The GCN encoder is updated only during critic training (backpropagated with the critic loss); it is frozen during actor updates.\n\n\n**Q7. Can DynSep handle nonlinear or MINLP problems?**\n\n**A7.**\n\nThank you for the insightful comment. We believe **DynSep can naturally extend to MINLPs** because SCIP’s unified Constraint Integer Programming (CIP) framework handles both MILPs and MINLPs within the same branch‑cut‑and‑price architecture. When solving MINLPs, SCIP automatically applies nonlinear handlers and MINLP-specific separators. like *RLT cut separator* and *minor cut separator*, to strengthen relaxations via valid cuts. \n\nBy incorporating MINLP‑specific separators into the action space, our instance‑aware configuration method could extend seamlessly to nonlinear problems. Meanwhile, we could enrich our state representation to capture nonlinear structure following the existing approach [2]. In other words, the core decision‑making paradigm—choosing which separators to fire at each node and when to stop—remains valid for both MILP and MINLP, making this extension both straightforward and feasible.\n\n\n**Q8. How does the blocked positional encoding impact performance?**\n\n**Q9. Would alternative architectures like graph transformers improve separator modeling?**\n\n**A8. & A9.**\n\nWe merged the results for **Q8** and **Q9** into a single table for compactness and easier comparison.\n\n+ **w/o BlockPE (for Q8):** This row show the results that replace the blocked positional encoding with a standard token-level positional encoding. Its **degraded performance relative to DynSep** indicates that masking intra-block order information (i.e., using BlockPE) benefits training and improves final performance.\n+ **GT (for Q9):**  This row show the results that replace the encoder’s GCN with a custom bipartite **graph transformer (GT)**: a multi-head TransformerConv for edge-aware message passing, followed by residual-connected LayerNorm and a two-layer feed-forward block. **\"GT\" shows no consistent improvement over DynSep**, which may be due to increased inference overhead or the need for finer stability/tuning to realize gains from the transformer-style aggregation.\n\n|Instance|Knapsack|Knapsack|Corlat|Corlat|neos|neos|supportcase|supportcase|\n|-|-|-|-|-|-|-|-|-|\n|Metric|Time(s)|PD integral|Time(s)|PD integral|Time(s)|PD integral|Time(s)|PD integral|\n|w/o BlockPE|0.93|11.12|34.4|2870.01|242.39|**8291.49**|153.29|12715.64|\n|GT|0.71|11.02|46.26|4563.26|243.52|12134.57|150.38|**9157.61**|\n|DynSep|**0.52**|**9.71**|**22.96**|**2233.42**|**235.19**|8511.58|**132.50**|9212.24|\n\n\n**Q10. Typo in Ablation Study.**\n\n**A10.** \n\nThank you for pointing this out. We will keep all statements consistent.\n\n\n[1] Huang, et al. \"Distributional MIPLIB: a multi-domain library for advancing ml-guided milp methods.\" arXiv 2024.\n\n[2] Tang, Bo, Elias Boutros Khalil, and Jan Drgona. \"Learning to Optimize for Mixed-Integer Nonlinear Programming.\" (2024)."}, {"review_id": "kiUULFi1qL", "reviewer": "Reviewer_rt7i", "summary": "This paper introduces DynSep, a dynamic separator configuration method for cutting plane selection in Mixed-Integer Linear Programming (MILP) solvers. DynSep formulates the round-wise activation and deactivation of cutting plane separators as a reinforcement learning (RL) problem. By modeling the evolving MILP instance as an incremental triplet graph, DynSep uses a GCN encoder and a decoder-only Transformer to autoregressively predict both when to halt separation and which separators to activate at each round. Experiments on synthetic and large-scale real-world MILP benchmarks demonstrate that DynSep significantly accelerates solving time (by 64% on average for easy/medium datasets) and reduces the primal-dual gap integral (by 16% on challenging datasets) compared to both human-designed and learning-based baselines. The method also shows strong generalization to much larger MILP instances than those used in training.", "questions": "- Could you add comparisons with these mainstream commercial solvers (such as Gurobi or CPLEX) so I can better understand the current level of DynSep’s capabilities in an industrial context?\n- How does DynSep handle the addition of new, user-defined, or more advanced separators beyond SCIP’s built-in set? What is required to support extensibility?", "limitations": "Yes", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 4, "significance": 4, "originality": 3, "strengths_and_weaknesses": "Strengths:\n- DynSep delivers substantial improvements over both human-designed and ML baselines across a wide range of MILP benchmarks, including large-scale industrial and public datasets.\n- The method generalizes to larger instances and unseen problem sizes, a key requirement for practical MILP solving.\n- Ablation studies confirm the necessity of each component, and visualization of learned separator policies aligns with domain insights.\n\nWeaknesses:\n- All experiments are conducted within SCIP; comparisons to commercial solvers (e.g., Gurobi, CPLEX) or other solver backends are missing, which limits claims of generality and practical impact.\n- The approach configures only the built-in separators in SCIP, and does not explore integration with advanced or user-defined cutting plane families, potentially limiting extensibility.", "was_revised": true, "final_justification": "Thank you for your detailed response. It has clarified the questions I raised and resolved my earlier concerns. Given this, I am confident in the contributions of the work and maintain my positive score of 4.", "rebuttal": "We thank the reviewer for the positive and insightful comments. We respond to each comment as follows and sincerely hope that our rebuttal will properly address your concerns. If so, we would deeply appreciate it if you could raise your score. If not, please let us know your further concerns, and we will continue actively responding to your comments and improving our submission.\n\n**Q1. Extension to commercial solvers. (e.g., Gurobi, CPLEX)**\n\n**A1.** \n\nWe appreciate the reviewer’s suggestion to apply our method to commercial solvers like Gurobi or CPLEX. Although our current experiments are conducted in SCIP, **translating our approach for separator-activation tuning to Gurobi is technically feasible.** Below, we outline the feasibility and limitations of such an implementation.\n\n+ **Feasibility.** Gurobi provides separators like `CliqueCuts`, `FlowCoverCuts`, `MIRCuts`, `StrongCGCuts`, etc., each accepting a configuration parameter in {0, 1, 2} (or –1 for automatic). 0 disables the cut separator, 1 enables it moderately, and 2 enables it aggressively—precisely matching DynSep’s configuration schema. These parameters can be set statically via `model.setParam(\"CliqueCuts\", value)` and can also be changed dynamically at the node level via `cbSetIntParam(...)` within a `where == GRB.Callback.MIPNODE` callback. By using these interfaces, our separator configuration method can be ported to Gurobi fairly directly.\n+ **Limitation.** Gurobi’s callback mechanism only invokes user code once per node for separators, which prevents us from adjusting parameters at a finer, per-separation-round granularity. However, node-level tuning remains effective for most practical scenarios.\n\nGiven the preceding discussion, the adaptation **requires substantial solver-specific engineering** to rewire state collection, accommodate Gurobi’s callback semantics, and translate our dynamic decision logic to its API. Due to time constraints, we have not yet completed the Gurobi experiments. Nevertheless, we remain confident that DynSep’s separator-configuration strategy can be properly implemented in Gurobi. We will **prioritize these tests** and include preliminary Gurobi results once feasible.\n\n\n\n**Q2. Integration with advanced or user-defined cutting plane families.**\n\n**A2.** \n\nSCIP exposes a **plugin API** for **user-defined separators**: once a custom separator is registered in SCIP, it appears in the separator nodes of our triplet graph alongside the other built-in separators. DynSep treats every separator node uniformly—built-in or custom—so any new separator simply becomes an additional token whose activation status can be tuned. In fact, our own “configuration separator” is implemented via this same API, and is invoked ahead of all other separators to adjust their parameters dynamically."}, {"review_id": "C5LeKyHQ99", "reviewer": "Reviewer_E4cm", "summary": "This paper proposes DynSep, a dynamic separator configuration method that models separator configuration in different rounds as a reinforcement learning task and tokenizes the incremental subgraphs with a transformer policy model to autoregressively predict when to half separation and which separators to activate at each round. Evaluated on synthetic and large-scale real-world MILP problems, DynSep substantially accelerates the solving time on easy and medium instances, as well as reducing integrality gap on hard instances.", "questions": "See weaknesses.", "limitations": "The authors explain the limitation of their methods in the conclusion section “the current implementation of dynsep lacks a lightweight decision model to retrain historical information across the global B&B tree”. I wonder if the authors can slightly expand in the conclusion to propose a few potential ways that can potentially address this limitation?", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 4, "significance": 3, "originality": 3, "strengths_and_weaknesses": "**Strengths**\n\n1. The paper is clear and well-written.\n2. The extension from static separator configuration to dynamic separator configuration is non-trivial. The authors handle this extension with incremental subgraph tokenization with blocked positional encoding for the transformer architecture, and these special treatments in the architecture design makes sense.\n3. The authors show significant performance improvement over baselines on a variety of benchmarks, with detailed ablation study to justify the method’s design choices.\n\n**Weaknesses**\n\n1. I feel like the methods developed in this paper is not particularly specific to configuring cutting plane separators, and it should be able to apply to configure other parameters (e.g. branching, heuristics). Can the authors comment on the difficulty / flexibility of adapting their framework to configure other parameters, and potentially do small experiments to show the feasibility of this extension? \n2. The authors set separator frequency as 10 and set the maximum separator round as T = 5. It is unclear how these two parameters (i.e. more or less frequent separator configurations) affect the performance of the method. The authors should perform some robustness studies in terms of varying these two parameters and comparing the performance.\n3. Can the authors conduct experiments on gurobi (or other MILP solvers) and see if their algorithms can improve gurobi as well?", "was_revised": true, "final_justification": "I'm overall in favor of accepting this paper, as it extends prior work from single-turn cutting-plane hyperparameter configuration to adaptive, multi-turn hyperparameter configuration, and it observes significant improvement from additional configurations. From the rebuttal, one weakness of the paper is that their current implementation cannot easily adapt to configure other sets of hyperparameters in the MILP solver, so the study of the proposed method's generalizability to different sets of MILP hyperparameters is bottlenecked by the engineering challenges. Also, the paper focuses on configuring the SCIP solver, and faces engineering challenges to adapt to other MILP solvers, so the generalizability across solvers is not tested. Given this, I would like to keep my initial score 4, borderline accept.", "rebuttal": "We thank the reviewer for the positive and insightful comments. We respond to each comment as follows and sincerely hope that our rebuttal will properly address your concerns. If so, we would deeply appreciate it if you could raise your score. If not, please let us know your further concerns, and we will continue actively responding to your comments and improving our submission.\n\n**Q1. The difficulty / flexibility of adapting their framework to configure other parameters**.\n\n**A1.** \n\nThank you for the insightful comment. **We claim that our proposed DynSep framework is inherently extensible to a broader set of solver hyperparameters beyond separators (e.g., branching, heuristics).** This is because DynSep is inherently a node-level adaptive configuration framework in the B&C tree. \n\nCompared to configure separator, adapting DynSep to branching or heuristics requires **nontrivial solver-specific engineering** because they expose different plugin/callback interfaces. Specifically, adapting to branching or heuristics would involve:\n\n1. Implement analogous instrumentation via the BRANCH and HEURISTIC plugin callbacks, such as `BRANCHEXECLP` and `HEUREXEC`.\n2. Augment the state representation with relevant branching/heuristic signals provided by the solver.\n3. Adapt the policy network of DynSep: model the outputs of other parameters as a logistic‑normal distribution, followed by optional discretization to support both integer and continuous parameters. This enables flexible, differentiable control of arbitrary solver parameters.\n4. Train the policy with the same PPO-based scheme. \n\nDue to time constraints, we have not yet migrated DynSep to branching or heuristic configuration, **but we have already extended it to additional cut-related parameters to demonstrate feasibility**. We conducted additional experiments on three critical hyperparameter groups, while retaining the original tuning mechanism for separator activation and max round.\n\n+ **para group 1: Cut Depth / Aggressiveness (sepastore/age in SCIP).** Controlled by solver parameters `separating/cutagelimit` and `separating/poolfreq`.\n+ **para group 2: Cut selection thresholds (e.g., efficacy vs. orthogonality).** Controlled by solver parameters `separating/minefficacy` and `cutselection/hybrid/minortho`.\n+ **para group 3: Separation frequency per node.** Controlled by solver parameters `separating/cutagelimit` and `separating/poolfreq`.\n\nWe provide the experimental results as follows.\n\n|Instance|Knapsack|Knapsack|Knapsack|Corlat|Corlat|Corlat|Supportcase|Supportcase|Supportcase|\n|-|-|-|-|-|-|-|-|-|-|\n|Metric|Time(s)|PD integral|PD gap|Time(s)|PD integral|PD gap|Time(s)|PD integral|PD gap|\n|DynSep (Ours)|0.52|9.71|0|**22.96**|2233.42|**0**|**132.50**|9212.24|7.86|\n|Para Group 1|0.65|9.01|0|47.36|4580.9| 4e+18  |141.54|8610.24|0.17|\n|Para Group 2|**0.36**|**7.91**|0|42.17|1971.18|0.01|167.64|8661.57|**0.16**|\n|Para Group 3|0.65|10.03|0|24.71|**1922.77**|0|141.09|**8362.53**|0.17|\n\nWe evaluated our method on three benchmark datasets: **Knapsack**, **Corlat**, and **MIPLIB mixed supportcase**, measuring solving time (*Time*), primal–dual gap integral (*PD integral*), and primal–dual gap (*PD gap*). All three metrics are *lower-is-better*.\n\nWe compared our original DynSep method with its extended versions incorporating three additional parameter groups. Results show that:\n\n1. Across all datasets, **DynSep and its extended variants consistently outperform the default solver configuration** reported in the main paper.\n\n2. The differences among the parameter groups are relatively small, yet certain combinations yield **further performance improvements** in both Time and PD integral.\n\nThese results validate that DynSep can flexibly **extend to control a broader set of solver hyperparameters**. Furthermore, carefully chosen parameter combinations can yield **additional gains** in solving speed and convergence.\n\n\n\n**Q2. Robustness studies in terms of separator frequency and maximum separation round.**\n\n**A2.** \n\nWe conducted robustness ablations over separator **frequency** (1, 5, 10, 20) and **maximum separation rounds** (3, 5, 10, 20) on three benchmarks (MIK, Corlat, and MIPLIB mixed Neos). Overall, **across almost all tested settings, DynSep outperforms the default configuration**, showing that the approach is reasonably stable to these hyperparameters. \n\nOn each benchmark, we report three metrics: solving time (*Time*), primal–dual gap integral (*PD integral*), and primal–dual gap (*PD gap*). All three metrics are *lower-is-better*. Below is our key observations.\n\n+ **Frequency:** Moderate frequency (e.g., 5&10) gives the better trade-off. That is, small frequency causes separators to fire excessively, incurring high cut-generation overhead, whereas large frequency reduces opportunities for timely dual-bound tightening.\n\n|Instance|MIK|MIK|MIK|Corlat|Corlat|Corlat|Neos|Neos|Neos|\n|-|-|-|-|-|-|-|-|-|-|\n|Metric|Time(s)|PD integral|PD gap|Time(s)|PD integral|PD gap|Time(s)|PD integral|PD gap|\n|Default|16.65|82.80|0.0|111.55|10573.14|2.73e+19|282.98|18500.5|2.5e+19|\n|Freq=1|12.74|124.22|0.0|53.77|4582.15|4e+18|252.12|8756.3|2.5e+19|\n|Freq=5|13.04|**86.55**|0.0|38.44|3513.13|0.0|243.91|9248.52|2.5e+19|\n|Freq=10 (Ours)|**10.99**|134.15|0.0|**22.96**|**2233.42**|**0.0**|**235.19**|**8511.58**|2.5e+19|\n|Freq=20|12.83|213.15|0.0|49.14|4658.09|0.0|261.86|8789.54|2.5e+19|\n\n+ **Maximum separation rounds:** Setting this value too low produces weak cuts and degrades performance, while setting it excessively high increases the computational cost of cut generation. Although this parameter shows some dataset sensitivity, MaxRound=5 is empirically near-optimal in our tests.\n\n|Instance|MIK|MIK|MIK|Corlat|Corlat|Corlat|Neos|Neos|Neos|\n|-|-|-|-|-|-|-|-|-|-|\n|Metric|Time(s)|PD integral|PD gap|Time(s)|PD integral|PD gap|Time(s)|PD integral|PD gap|\n|Default|16.65| 82.80       | 0.0    | 111.55    | 10573.14    | 2.73e+19 | 282.98     | 18500.5     |2.5e+19|\n|MaxRound=3|13.63|150.19|0.0|107.26|8965.26|2.4e+19|239.22|8581.5|2.5e+19|\n|MaxRound=5 (Ours)|**10.99**| **134.15**  | 0.0    | **22.96** | **2233.42** | 0.0      | **235.19** | **8511.58** |2.5e+19|\n|MaxRound=10|17.6|182.71|0.0|40.45|3717.64|2e+18|249.41|8846.91|2.5e+19|\n|MaxRound=20|16.2|169.8|0.0|36.71|3609.11|0.0|252.59|10945.82|2.5e+19|\n\n\n\n**Q3. Can the authors conduct experiments on gurobi (or other MILP solvers)?**\n\n**A3.**\n\nWe appreciate the reviewer’s suggestion to apply our method to ohter MILP solvers like Gurobi.  Although our current experiments are conducted in SCIP, **translating our approach for separator-activation tuning to Gurobi is technically feasible.** Below, we outline the feasibility and limitations of such an implementation.\n\n+ **Feasibility.** Gurobi provides separators like `CliqueCuts`, `FlowCoverCuts`, `MIRCuts`, `StrongCGCuts`, etc., each accepting a configuration parameter in {0, 1, 2} (or –1 for automatic). 0 disables the cut separator, 1 enables it moderately, and 2 enables it aggressively—precisely matching DynSep’s configuration schema. These parameters can be set statically via `model.setParam(\"CliqueCuts\", value)` and can also be changed dynamically at the node level via `cbSetIntParam(...)` within a `where == GRB.Callback.MIPNODE` callback. By using these interfaces, our separator configuration method can be ported to Gurobi fairly directly.\n+ **Limitation.** Gurobi’s callback mechanism only invokes user code once per node for separators, which prevents us from adjusting parameters at a finer, per-separation-round granularity. However, node-level tuning remains effective for most practical scenarios.\n\nGiven the preceding discussion, the adaptation **requires substantial solver-specific engineering** to rewire state collection, accommodate Gurobi’s callback semantics, and translate our dynamic decision logic to its API. Due to time constraints, we have not yet completed the Gurobi experiments. Nevertheless, we remain confident that DynSep’s separator-configuration strategy can be properly implemented in Gurobi. We will **prioritize these tests** and include preliminary Gurobi results once feasible.\n\n\n\n**Q4. A lightweight decision model to retrain historical information across the global B&B tree.**\n\n**A4.**\n\nHere we provide more discussion on the potential ways that can address the mentioned limitation. The current DynSep only reasons over the **local, per-node** incremental graph. To capture **global, historical dynamics** across the entire branch-and-bound tree, one could:\n\n- **Maintain a tree-wide memory** of separator performance and variable bound changes (e.g. via a lightweight recurrent summary or graph coarsening).\n- **Aggregate cut efficacy statistics** at each depth level (or over recent sibling nodes) to inform decisions in new nodes—trading off memory overhead against improved long-term planning.\n\nWhile logging every cut and bound update could incur runtime overhead, designing rational sampling techniques could keep this overhead minimal. Embedding such a lightweight global module would allow DynSep to adapt not only to per-round shifts but also to the evolving solver trajectory across the entire search tree."}, {"review_id": "z2x80P3QyG", "reviewer": "Reviewer_Fjx6", "summary": "The paper proposes DynSep, a reinforcement learning-based method for dynamically configuring cutting plane separators in mixed-integer linear programming (MILP) solvers.", "questions": "(1) Could you specify which instances you used to test the performance in MIPLIB. What do you mean by mixed neos and mixed suportcase specifically?\n(2) For real-world use, what is the latency of policy inference during B&C?", "limitations": "(1) DynSep ​only optimizes separator activation status​ (+1, 0, -1) and round termination (m_t). It ignores critical hyperparameters like:\nCut depth/aggressiveness (e.g., sepastore/age in SCIP); Cut selection thresholds (e.g., efficacy vs. orthogonality) and Separation frequency per node.\n\n(2) Memory/compute inference overhead is not discussed. Which scenarios and practical cases the data-driven approach is appropriate should be addressed.", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 2, "significance": 2, "originality": 2, "strengths_and_weaknesses": "The strength of this paper:\n\n(1) Triplet graphs are used to capture evolving problem states during branch and cut with Lightweight subgraph updates \n(2) Decoder-only design with blocked positional encoding captures temporal dependencies across rounds while ensuring permutation equivariance\n\nThe weakness of this paper:\n\n(1) limited scope of hyperparameter tuning (separator activation/timing)\n(2) limited scope of testing datasets (MIPLIB neos and supportcase)", "was_revised": true, "final_justification": "At the last minutes of rebuttal, the author provided more evaluation experiments and addressed my concerns.", "rebuttal": "We thank the reviewer for the insightful and valuable comments. We respond to your comments as follows and sincerely hope that our rebuttal could properly address your concerns. If so, we would deeply appreciate it if you could raise your score（\"3: Borderline Reject\"). If not, please let us know your further concerns, and we will continue actively responding to your comments and improving our submission.\n\n**Q1. Lmited scope of hyperparameter tuning, ignoring critical hyperparameters.**\n\n**A1.**\n\nThank you for the insightful comment. **We claim that our proposed DynSep framework is inherently extensible to a broader set of solver hyperparameters beyond separator activation and timing.** This is achieved by adapting the policy network: we model the outputs of additional parameters as a logistic‑normal distribution, followed by optional discretization to support both integer and continuous parameters. This enables flexible, differentiable control of arbitrary solver parameters.\n\nTo validate this, we conducted additional experiments on three critical hyperparameter groups (as suggested), while retaining the original tuning mechanism for separator activation and max round.\n\n+ **para group 1: Cut Depth / Aggressiveness (sepastore/age in SCIP).** Controlled by solver parameters `separating/cutagelimit` and `separating/poolfreq`.\n+ **para group 2: Cut selection thresholds (e.g., efficacy vs. orthogonality).** Controlled by solver parameters `separating/minefficacy` and `cutselection/hybrid/minortho`.\n+ **para group 3: Separation frequency per node.** Controlled by solver parameters `separating/cutagelimit` and `separating/poolfreq`.\n\nWe provide the experimental results as follows.\n\n|Instance|Knapsack|Knapsack|Knapsack|Corlat|Corlat|Corlat|Supportcase|Supportcase|Supportcase|\n|-|-|-|-|-|-|-|-|-|-|\n|Metric|Time(s)|PD integral|PD gap|Time(s)|PD integral|PD gap|Time(s)|PD integral|PD gap|\n|DynSep (Ours)|0.52|9.71|0|**22.96**|2233.42|**0**|**132.50**|9212.24|7.86|\n|Para Group 1|0.65|9.01|0|47.36|4580.9| 4e+18  |141.54|8610.24|0.17|\n|Para Group 2|**0.36**|**7.91**|0|42.17|1971.18|0.01|167.64|8661.57|**0.16**|\n|Para Group 3|0.65|10.03|0|24.71|**1922.77**|0|141.09|**8362.53**|0.17|\n\nWe evaluated our method on three benchmark datasets: **Knapsack**, **Corlat**, and **MIPLIB mixed supportcase**, measuring solving time (*Time*), primal–dual gap integral (*PD integral*), and primal–dual gap (*PD gap*). All three metrics are *lower-is-better*.\n\nWe compared our original DynSep method with its extended versions incorporating three additional parameter groups. Results show that:\n\n1. Across all datasets, **DynSep and its extended variants consistently outperform the default solver configuration** reported in the main paper.\n\n2. The differences among the parameter groups are relatively small, yet certain combinations yield **further performance improvements** in both Time and PD integral.\n\nThese results validate that DynSep can flexibly **extend to control a broader set of solver hyperparameters**. Furthermore, carefully chosen parameter combinations can yield **additional gains** in solving speed and convergence.\n\n\n**Q2. Limited scope of testing datasets (MIPLIB neos and supportcase)；**\n\n**A2.** \n\nThank you for raising this point. We **have extended our evaluation beyond MIPLIB mixed neos and mixed supportcase**, including two real-world datasets from the Distributional MIPLIB benchmark [1]:\n\n+ **Maritime Inventory Routing Problem (MIRP).** MIRP arises in bulk shipping logistics, integrating vessel routing and port inventory decisions, featuring 15080 binary variables, 19576 continuous variables, and 44430 constraints.\n+ **Seismic‑Resilient Pipe Network Planning (SRPN).** SRPN involves optimizing municipal water pipe network design to ensure resilience under seismic disturbances, featuring 3016 binary variables, 3016 continuous variables, and 5917 constraints.\n\nWe set the time limit to $600$ seconds. The results are summarized in the table below. \n\n|Instance|mirp|mirp|mirp|srpn|srpn|srpn|\n|-|-|-|-|-|-|-|\n|Metric|Time(s)|PD integral|PD gap|Time(s)|PD integral|PD gap|\n|NoCuts|486.68|34735.55|6.67e+18|**280.7**|11020.69|0.28|\n|Default|580.98|52362.46|5.38e+19|332.04|11687.01|0.21|\n|Search(20)|492.85|33028.81|6.30e+18|384.02|14571.56|0.26|\n|Prune|501.49|35542.82|8.33e+18|296.08|9211.76|0.19|\n|LLM4sepasel|511.97|34573.52|5e+18|300.66|8421.65|0.14|\n|DynSep(Ours)|**482.13**|**30838.39**|**1.38**|294.77|**7581.16**|**0.1**|\n\nThese results show that our method (**DynSep**) delivers **marked performance gains** on additional real‑world datasets (MIRP and SRPN). Compared to the other configuration methods, DynSep significantly improves both **convergence speed** (evidenced by reduced PD integral) and **solution quality** (evidenced by lower PD gap).\n\n\n**Q3. Specification of MIPLIB mixed neos and mixed supportcase.**\n\n**A3.** \n\nWe apologize for the confusion. *MIPLIB mixed neos* and *mixed supportcase* evaluated in our paper are two well-established subsets derived from MIPLIB 2017 [2], which were constructed by the prior learning-based MILP research [3].\n\n**Each subset comprises a cluster of similar instances selected from MIPLIB 2017, where similarity is quantified using 100 human-designed instance features [1]**. Briefly, a seed instance is first selected: *neos‑1456979* for the mixed neos set, representing knapsack‑constraint problems, and *supportcase40* for the mixed supportcase set, representing set‑packing problems. Then, a data-driven clustering algorithm [3] is applied to gather MIPLIB instances that closely match the seed instance in feature space, constructing the mixed neos and mixed supportcase subsets.\n\nDue to **space limitations** in the rebuttal, we **only present a subset of** MIPLIB mixed neos and mixed supportcase datasets used in our experiments:\n\n+ Mixed neos contains 17 instances: neos-1456979, icir97 tension, etc.\n\n+ Mixed supportcase contains 37 instances: supportcase40, acc-tight2, acc-tight4, etc.\n\n\n**Q4. The latency of policy inference during B&C.**\n\n**A4.** \n\nThank you for raising this point. We have provided **per-decision latency** (the latency of policy inference per decision step) and the **total inference time** through the solving process as follows.\n\n+ **Per-decision latency**: The table below reports the **average** (\"Avg. latency\") and **worst-case** (\"Max. latency\") time taken for a single policy call across the entire branch-and-cut process. \n\n||Set Cover|MIS|Knapsack|Corlat|MIK|Anonymous|LB|Neos|Supportcase|\n|-|-|-|-|-|-|-|-|-|-|\n|Avg. Latency (s)|0.33|0.22|0.09|0.05|0.31|0.41|3.04|0.11|0.39|\n|Max. Latency (s)|1.09|1.01|0.71|0.70|1.07|1.94|4.51|2.02|6.85|\n\nOur results reveal that **per-decision latency increases as instance size grows**. The **average inference latency** stays below 0.5 seconds across all datasets, with the exception of the largest load balancing (LB) instance.\n\n---\n\nThe two tables below provide the **total inference time** (\"Infer. Time\") over different datasets and solver time limits, along with the **inference overhead rate** (\"Overhead Rate\"), which represents the percentage of policy inference in the total solving runtime.\n\n+ **Total inference time (nine datasets in our manuscript, $300$-second time limit):**\n\n||SetCover|MIS|Knapsack|Corlat|MIK|Anonymous|LB|Neos|Supportcase|\n|-|-|-|-|-|-|-|-|-|-|\n|Infer. Time (s)|1.05| 0.41  |0.42|16.56| 1.57  |14.73|10.54| 23.9 |14.16|\n|Overhead Rate (%)|69.54|77.36|80.77|72.13|14.29|6.09|3.51|10.16|10.69|\n\n+ **Total inference time (hard datasets in our manuscript, $3600$-second time limit):**\n\n||Anonymous|LB|Neos|Supportcase|\n|-|-|-|-|-|\n|Infer. Time (s)|21.32|16.01|86.96|12.31|\n|Overhead Rate (%)|0.89|0.44|3.19|2.17|\n\nThese tables show that on larger, more complex instances and with longer solver time limits, **a smaller share** of the total solving time is spent on decision making. This implies that **the computational cost of our policy calls becomes more justifiable at scale**, enabling meaningful improvements in overall solver efficiency with only modest overhead investment.\n\n\n**Q5. Memory/compute inference overhead is not discussed.**\n\n**A5.** \n\nWe have analyzed the compute inference overhead in Q4. Here, we focus on the memory inference overhead.\n\n+ **Per-decision memory overhead:** The table below show the peak GPU memory usage per policy call.\n\n||SetCover|MIS|Corlat|LB|Neos|\n|-|-|-|-|-|-|\n|Max. CPU mem. (MB)|3081.70|2877.39|2892.15|4852.27|3100.90|\n|Max. GPU mem. (MB)|2.10|2.10|2.11|2.11|2.11|\n\n+ **Peak memory overhead for overall inference:**\n\nThe memory footprint required to store the encoder and policy model weights is $2.08$ MB for each dataset. We track the peak GPU memories during the reference via the tool `torch.cuda.max_memory_allocated()`, which are list in the table below. \n\n|| SetCover |MIS|Corlat|LB|Neos|\n|-|-|-|-|-|-|\n|Peak GPU mem. (MB)|35.56|57.85|15.68|257.60|28.85|\n\n\n**Q6. Which scenarios and practical cases the data-driven approach is appropriatML-guided MILPe should be addressed.**\n\n**A6.**\n\nOur data‑driven configuration approach proves effective across both **synthetic** and **real‑world** MILP benchmarks [1,2,3]. It is particularly well‑suited to scenarios where one has access to a batch of instances drawn from the **similar distribution** as the unseen examples. In these cases, the policy network can be trained on those example instances and then deployed during branch‑and‑cut to flexibly tune solver parameters via the solver’s existing parameter interfaces, dynamically adapting to individual problem structure. \n\n[1] Huang, et al. \"Distributional MIPLIB: a multi-domain library for advancing ml-guided milp methods.\" arXiv 2024.\n\n[2] Gleixner, et al. \"MIPLIB 2017: data-driven compilation of the 6th mixed-integer programming library.\" Mathematical Programming Computation(2021).\n\n[3] Wangi, et al. \"Learning cut selection for mixed-integer linear programming via hierarchical sequence model.\"ICLR (2023)."}]

Author rebuttal:

Meta-review: This paper introduces DynSep, a reinforcement learning-based framework aimed at enhancing a backbone MILP solver through dynamic separator configuration. The approach integrates an incremental subgraph-based state representation into a transformer policy model, which autoregressively predicts both when to terminate separation and which separators to activate during branch-and-cut. Extensive empirical evaluation demonstrates that the method significantly accelerates the SCIP solver across diverse datasets, while also exhibiting strong generalization to unseen MILP instances.

There is a clear consensus among the reviewers that the proposed method is technically sound, effective, and delivers notable performance improvements. The extensive experimental results, generalization capability, and the inclusion of interpretability analysis are all strengths of the paper. Most of the key claims are well substantiated through ablation studies. No critical flaws are found. However, the main limitation lies in the lack of study on the broader applicability of the method, particularly its applicability to commercial solvers such as Gurobi, and a wider range of branch-and-cut parameters. A further concern is the limited discussion on the effectiveness of the incremental triplet graph compared to full-graph encoding. While the paper attributes its use to improved inference speed, ablation results show that it also improves performance. The authors are encouraged to revise the paper to include a more detailed analysis explaining this.

Despite these cons, the reviewers agree that the paper makes a meaningful contribution to the field. The authors' rebuttal addressed the technical and empirical concerns effectively, providing additional evaluations that further validate the approach. Following the discussion, all reviewers recommend acceptance. The required revisions are expected to be manageable in the camera-ready version. I am happy to recommend acceptance, provided that the camera-ready version includes the promised experiments and adequately addresses the above weaknesses.
Decision: Accept (poster)