FMIP: Joint Continuous-Integer Flow For Mixed-Integer Linear Programming
Authors: Hongpei Li
Hui Yuan
Han Zhang
Jianghao Lin
Dongdong Ge
Mengdi Wang
Yinyu Ye
Venue: ICLR 2026 Poster (2026)
Keywords: Mixed Integer-Linear Programming
Learning to Optimize
Flow Matching
Primary area: optimization
TL;DR: FMIP is a generative framework that jointly models integer and continuous variables in MILP, achieving a 41.34% reduction in primal gap and demonstrating compatibility with various solvers and applications.

Abstract: Mixed-Integer Linear Programming (MILP) is a foundational tool for complex decision-making problems. 
However, the NP-hard nature of MILP presents a significant computational challenge, motivating the development of machine learning-based heuristic solutions to accelerate downstream solvers. 
While recent generative models have shown promise in learning powerful heuristics, they suffer from a critical limitation.
That is, they model the distribution of only the integer variables and fail to capture the intricate coupling between integer and continuous variables, creating an information bottleneck and ultimately leading to suboptimal solutions.
To this end, we propose Joint Continuous-Integer Flow for Mixed-Integer Linear Programming (FMIP), which is the first generative framework that models the joint distribution of both integer and continuous variables for MILP solutions.
Built upon the joint modeling paradigm, a holistic guidance mechanism is designed to steer the generative trajectory, actively refining solutions toward optimality and feasibility during the inference process. 
Extensive experiments on eight standard MILP benchmarks demonstrate the superior performance of FMIP against existing baselines, reducing the primal gap by 41.34% on average. 
Moreover, we show that FMIP is fully compatible with arbitrary backbone networks and various downstream solvers, making it well-suited for a broad range of real-world MILP applications.

Published as a conference paper at ICLR 2026
FMIP: JOINT CONTINUOUS-INTEGER FLOW FOR MIXED-INTEGER LINEAR PROGRAMMING
Hongpei Li', Hui Yuan², Han Zhang", Jianghao Lin Dongdong Ge .Mengdi Wang", Yinyu 1,4,56 Shanghai University of Finance and Economics, Shanghai, China
2 Princceoon University, New Jersey, USA
National University of Singapore, Singapore
4 Antai College of Economics and Management, Shanghai Jiao Tong University, Shanghai, China
5 SShnnghai Institute for Mathematics and Interdisciplinary Sciences, Shanghai, China
6 "Stanford University, California, USA
i shongpei 1i @gmail.... com, lin sinjiarnhhoodsjtu.
ABSTRACT
Mixed- Mixed-Integer Linear Programming (MILP) foundational tool for complex decision-making problems. However, the NP-hard nature of MILP presents a sig- nificant computational challenge, motivating the development of machine learning- based heuristic solutions to accelerate downstream solvers. While recent generative models have shown promise in learning powerful heuristics, they suffer from a critical limitation. That is, they model the distribution of only the integer variables and fail to capture the intricate coupling between integer and continuous variables, creating an information bottleneck and ultimately leading to suboptimal solutions. To this end, we propose Joint Continuous- Integer Flow for Mixed-Integer Linear Programming (FMIP), which the first generative framework that models the joint distribution of both integer and continuous variables for MILP solutions. Built upon the joint modeling paradigm, holistic guidance mechanism is designed steer the generative trajectory, actively refining solutions toward optimality and feasibility during the inference process. Extensive experiments on eight standard MILP benchmarks demonstrate the superior performance of FMIP against ex- isting baselines, reducing the primal gap by 41.34% on average. Moreover, we show that FMIP is fully compatible with arbitrary backbone networks and various downstream solvers, making it well-suited for broad range real-world MILP applications. Our code is available
INTRODUCTION
Mixed-Integer Linear Programming (MILP) represents a cornerstone mathematical optimization, providing a powerful framework for modeling complex decision-making problems that involve both discrete choices and continuous quantities /hhuu al. (2025); Kratica et al.) (2014); He al.) (2013). Its ability to capture the intricate interplay between integer and continuous variables makes indispensable across diverse domains, including combinatorial optimization (Della Croce and Paschos 2014), energy systems (Michling et al. 2023), and supply chain design (Ivanov et al. 2022).
Despite its versatility, solving a MILP instance remains a fundamental challenge due its NP-hard property (Bertsimas and Tsitsiklis 1997). Consequently, exact solvers often rely heuristics to find high-quality solutions in a practical amount of time. This has motivated a surge of interest in using deep learning predict powerful heuristics that can warm-start and guide the downstream solvers (Nair et al.) 2020) Han et al., 2023] Liu et al.) 2025 Chen et al. 2025). Among them, generative modeling has emerged
particularly promising approach. Modern techniques like diffusion models and flow matching reframe the difficult one-step task of predicting complete solution into iterative refinement process (Lin al. 2024b Sun and Yang 2023 Feng al.
2024. Zeng et al. 2024). By learning to transform a simple noise distribution into distribution of
"Jianghao Lin is the corresponding author. huts://github.com hongpei/FMiPP

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Published as a conference paper at ICLR 2026
Coupled Relationship Between Integer Continuous Variables
Existing Work
Integer
Costimoou
Predict
d
Calculate Guidance
?
Uptinal
min
d
Objectivve
1-1
3
Partial Distribution Modellin en Integor Variables
Sabootimal Guidance
d
Ee
2 3,
j-
Peastbllity
Our FMIP
Σ d
2 Σ
s 0,
Desination 0.0
0.5
Gundance
Calculate
1-1
Uptimal Putin
d: e (0,1),
= 2,2.3
1.0
10.1,
1,
Joint Distribution Modeling Integer Continuous Variables
Holistic Guidance
The key advantages of our FMIP over existing works: 1) the joint distribution modeling on both integer and continuous variables and 2) the consequent holistic guidance during inference.
Figure 1:
high-quality solutions, these models can effectively navigate the complex combinatorial search space, breaking down the challenge of satisfying intricate constraints into series of more manageable steps.
However, existing generative methods for MILP suffer from a critical limitation: they model the distribution of only the integer variables. As illustrated in Figure [ this architectural choice ignores the strong, coupled relationship between integer and continuous variables, which is fundamental to the structure of MILP problems. Since both variable types often required evaluate the objective function and check for constraint satisfaction, this incomplete modeling creates an information bottleneck, leading suboptimal solution quality and hindering the effectiveness of any guidance mechanism.
To this end, we propose Joint Continuous- Integer Flow for Mixed- Laxegernteger Linear Programming (FMIP), which is the first generative framework that models the joint distribution of both integer and continuous variables for MILP solutions. FMIP leverages a conditional flow matching process to progressively generate complete solution, fully capturing the interdependence between all decision variables. This joint modeling approach unlocks a holistic guidance mechanism that can steer the generation process using complete, instance- wise feedback from both the objective function and constraint violations. We highlight that while the transition from ILP to MILP appears intuitive, joint modeling represents critical but previously overlooked gap. Bridging this gap is crucial: it enables us to shift from single-poin prediction to generative distribution modeling, addressing the inherent "one-to-min fone-to-many" nature MILP solutions (multiple optima), and 2) enables the utilization of instance- wise optimization information, addressing a critical limitation of partial modeling schemes. FMIP fully compatible with arbitrary backbone networks and downstream solvers, showing 41.34% relative improvement on average across eight standard MILP benchmarks over SOTA baselines. Our contributions are summarized as follows:
We introduce FMIP, the first generative framework to jointly model the complete distribution of both integer and continuous variables for MILP solutions, effectively capturing their critical coupling relationship.
We design a holistic guidance mechanism that leverages the jointly generated variables to steer the sampling process toward solutions with better objective values and constraint satisfaction.
As powerful generative learning paradigm, FMIP is agnostic the choice of backbone networks and downstream solvers, making it well-suited for broad range of real-world MILP applications.
FMIP sets a new state-of state-of-the-arr -he-art for carning-based MILP heuristics, achieving superior results on eight benchmarks while showing compatibility with diverse backhones and solvers.
2 RELATED WORKS
Machine learning is widely used to accelerate Mixed Integer Linear Programming (MILP) solvers. One major line of research focuses enhancing the internal components of the Branch-and -adnchaand-Boud
algorithm, such as learning policies for variable branching (Gasse et al., 2019; Khalil et al. 2016
Gupta et al. 2020: Zarpellon et al. 2021 Gupta al. 2022 Scavuzzo al. 2022 Lin al. 2024a
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Zhang al. 2024) cutting plane selection (Tang al. 2020 Huang al. 2022). Another closely
related direction, which our work belongs to, aims to predict high-quality heuristic solutions to
warm-start the optimization process, thereby reducing the search space and speeding up convergence
(Nair et al. 2020 Han et al. 2023 Huang et al.) 2024b Liu et al. 2025 Chen et al. 2025 Zeng
al.
2024
solution-prediction prediction methods are discriminative in nature, employing models like Graph Neural Networks (GNNs) predict the variable assignments in a single forward pass (Huang et al.
Most existing solution-
Liu ct al. 2025 Hu et al. 2024) Huang et al. 2024a] Ye et al. 2024). However, predicting complete, feasible, and high-quality solution in one step is exceptionally challenging due the complex combinatorial structure of MILPs. This difficulty has motivated the use of generative models, such as diffusion models and flow matching, for MILP heuristics (Feng et al. 2024, Zeng al.. 2024). These models reframe the solution prediction as an iterative reñnement process, decomposing the hard single-step task into a more manageable multi-step generation, which better suited for navigating complex constrained spaces (Zeng et al., 2024). However, identify fundamental limitation in these state-of-the-art approaches: prior approaches generally focus on partial distribution modeling integer variables, which inherently precludes capturing the critical interdependence between all decision variables, leading to suboptimal guidance and inferior solutions. contrast, our proposed Joint Continuous-Integer Flow for MILP is the first generative framework that explicitly models the joint distribution of both continuous and integer variables, directly addressing this critical gap in the literature.
2024b
PRELIMINARIES
3.1 MILP DEFINITION AND GRAPH REPRESENTATION
Suppose we have n decision variables denoted as (æzz æee E 2 x R"-9, where øy and æe represent the integer and continuous variables, respectively. An MILP instance can be defined as:
min s
T æ
s.t.
Ar < b
1 S S u
(I)
vr E {0, (0,1..... K}
æc E 1R--
Here, w is the objective function coefficient vector. I and u are the lower bounds and upper bounds for decision variables. A E m and E Rm denote the coefficient matrix and the right-hand-side vector of linear constraints. Moreover, K is scalar integer defining the maximum feasible value of integer variables (i.c., x, E 1,, ............ variables can thus be represented by the tuple M (A, b, I, u, w). A solution æ is considered feasible if all constraints are satisfied. Note that, in this paper, we focus on MILP problems with bounded integer variables, most real-world applications exhibit this characteristic. Specially, when K = such that æх E {0, 133, the problem reduces to Mixed Integer Binary Programming (MIBP), which is the most commonly studied subclass of MILP (Huang al. 2024b Liu al., 2025). Our FMIP framework can naturally extend to unbounded integer variables by incorporating the bit-wise prediction strategy employed in prior studies (Nair et al.) 2020 Han al. 2023). Alternatively, we can continuously relax a subset of these variables to predict their continuous embeddings, which are subsequently discretized via rounding or projection to obtain valid integer assignments.
K}. .)). Vi E I). An MILP instance with m linear constraints and
Graph Representation for MILP. When applying deep learning to MILP, it is common to represent an MILP instance as bipartite graph (Gasse et al. [019). The bipartite graph consists of two sets of nodes: one representing the decision variables and the other representing the constraints. Edges in this graph connect variables to the constraints which they appear, effectively encoding the sparsity pattern of the constraint matrix. Building on this foundation, shown in the top-left part of Figure [] we maintain the fundamental bipartite topology but explicitly distinguish between integer and continuous variables as distinct node types. Specifically, an MILP instance is denoted as graph G = (Vz, Vc, Veen;& E), where the nodes are partitioned into three distinct sets:
Integer Variable Nodes (Vz): One node for each of the q integer variables.
Continuous Variable Nodes (Vc): One node for each of the q continuous variables.
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Constraint Nodes (Voo)): One node for each of the linear constraints.
An edge is drawn between variable node j E Vr U Ve and constraint node E Veon if the variable Ty has a non-zero coefficient in the i-th constraint. Based on the rich, relational MLIP graph, various graph neural networks (e.g., GCN (Kipf 2016) or GAT (Veličković et al. (017)) can be employed to extract the rich topological representations for later discriminative or generative learning processes. More details can be found in Appendix B and Appendix C
3.2 FLOW MATCHING
Flow Matching (FM) is powerful framework for training generative models (Lipman et al. 2023
Albergo and Vanden- Eijndenn 2023 Liu et al., 2023). The core idea to learn time-dependent vector field tú that transforms samples from a simple prior distribution Po (e.g.. Gaussian noise) into samples from target data distribution p1. This transformation is defined by a probability path (Pt)te(o.1) interpolating between Po and pl, which is induced by an Ordinary Differential Equation (ODE): de, (c,)). FM allows the vector field v, be learned directly via simple regression objective, given defined path between noise and data samples. FM is flexible and can model both continuous and discrete data, which is essential for our work.
For continuous data, the prior Po is typically a standard Gaussian N(,0 1). The interpolating path between data point ej and noise is defined c, c, (tc1,, (1 I). This yields simple, closed-form target vector field for model training, i.e., vr(cel(1) = -==
For discrete data, the prior Po a uniform categorical distribution over categories. The condi- tional path for each component i of a data point di is defined as di { ~ Cat(to(d,' (d())=
(1 t)/ K), where 8 the Kronecker delta and Cat(-) denotes the categorical distribution. This path implies a target conditional rate matrix R (l())) for model training:
.3) 1-t (1- (1-6(d)).c) di.))
(
Id
(2)
During inference, these targets replaced by the model's predictions to generate new samples by evolving the state over time. At each step, the learned model provides an estimate of the vector field or rate matrix ft, which is used to update the current state:
c++ c. (c)(ecl))) () LAA Cat ( (()) la
for continuous data,
d d(ad)t). for discrete data.
4 FMIP: JOINT CONTINUOUS-
FLOW FOR MILP
In this section, introduce FMIP, a generative framework that learns the joint distribution of integer and continuous variables to find high-quality solutions for MILP instances. This unified modeling the cornerstone of our approach, explicitly designed capture variable coupling and enable the utilization of aistance-wise optimization information, which is inaccessible to partial modeling. As shown Figure 2 FMIP is built on a time-dependent graph representation, a joint continuous-integer flow model, a holistic guidance mechanism for inference, and seamless integration with downstream solvers. Notably, our FMIP framework is independent of the choice generative model. We employ flow matching in this work due its conceptual simplicity and training stability.
4.1 DEPENDENN GRAPH REPRESENTATION
FMIP operates on a time-dependent solution graph, which captures both the static structure of an MILP instance and its dynamic state during the generative process. As defined in Section 3.1 the static structure is encoded in a graph G (Vz. Ve, Veen:
During the flow matching process, define the solution graph at time-step E [0, 1] as the tuple G, (G, d,, c),, where d, and c are the values of the integer and continuous variables, respectively. This dynamic state information incorporated directly into the graph by augmenting the static features of each variable node. We define H; = (dc, c)) as the vector of solution values for all variables. x, denotes the state of the i-th variable time step t.
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Overall Framework of Solving MILP Based on Our FMIP
Var lables
1.0
FMIP
Downstrearn Solver
0.0
Ed-2
6.0
4 (.33).3)) 1.3.1.
Consiuucus rablei
1-122.
Graph (G)
MILP Solution
MILP Problem
Joint Continuous- Integer Flow for MILP
(di, G)
(d
MILP Graph
Predictor
(4 Cq)
(d)
(d,, 61)
Joint Multi-Ster Step Generation
Discrete State
Continuous State
Feasible Region
Real Step
Guidance
Feasibiity
Objecttre
Velcotty
Figure 2: The overall framework of our proposed FMIP.
4.2 JOINT CONTINUOUS INTEGER FLOW
To overcome limitations of integer-only models, we design a flow matching process over mixed solution space of both continuous and integer variables. The process constructs probability path Pij (Gr G1) for E [o, 1], which transforms a simple noise distribution po (o into the target distribution high-quality solutions p1 (G1).
Joint Training Objective Our model, parameterized by 8, learns this transformation by predicting the target solution (d1,c) from a noisy state G,. It features two prediction heads over a shared learnable graph backbone network (e.g., GCN GAT). One head outputs the denoised continuous values è (G.) and another outputs the probability distribution of the integer solution p(d The model is trained by minimizing a joint loss function that combines regression loss for the continuous variables and a cross-entropy loss for the integer variables:
|è (G:) ci
((0) = Eg
log ((d
1 1
where / ~~ ((00 1), 11 Pdata. and is hyperparameter balancing the terms.
Sampling Process At inference, we generate solution by simulating the forward ODE from t = 0 t At each time step f, the model takes the current noisy graph G, as input and produces its predictions, i.e., Ci (G.) and p(d, Gt). As illustrated in Eq. 3 these outputs are used to estimate the conditional vector field v, and rate matrix Rej1- enabling the next step of the simulation. We use cosine schedule for time discretization, which allocates more steps to the low-noise region near f = 1 for better quality (Nichol and Dhariwal 202I). This generative formulation a key advantage over traditional one-step discriminative predietors, as it enables the multi-step iterative refinement with holistic guidance mechanism during sampling, which we detail next.
4.3 HOLISTIC GUIDANCE MECHANISM
A key benefit of FMIP is that the model predicts complete solution candidate (d, é) at any step t during generation. Hence, holistic guidance can be directly derived using instance-specific information from the MILP formulation itself (both the objective function and constraints), which steers the generation process towards solutions that are both feasible and optimal. Such capability is absent in previous works that only consider integer variables.
Guidance Target Function. We define a target function f fxx that captures the two primary goals of MILP P solution: minimizing the objective value and the constraint violation. The function is
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weighted sum of the objective and a penalty for constraint violations:
s(d, = [(x) = w x + [max(0, Ar, -
(S)
where æ = (d, c) is the set of all variables, is the weight of constraint-violation term to balance two guidance parts, and A.,. the i-th row of the constraint matrix. The model is guided to generate solutions that minimize this function. Following Lin et al. (202), we implement guidance by modifying the update steps for the continuous and integer variables separately.
Guidance on Continuous Variables. For the continuous variables, we employ gradient- gradient-based based guidance. At each step i, we perform gradient descent on the target function f w.r.t the predicted con- tinuous values è (Gz), steering them towards regions of lower objective value and higher feasibility:
C++ Project,, (ct - Pc Ve. ((()))
(6)
where di is sampled from the predicted integer distribution pidd (.)) and is the step size. The projection function Project,,,,) constrains the continuous variables within their specified bounds.
Guidance Integer Variables. For the integer variables, adopt an effective sampling and- reweighting scheme motivated by recent work (Lin et al.) (025). Instead of computing gradients, at time step t. sample batch of B candidate integer solutions {dije,r)," } from the model's current predicted integer distribution (()) We then reweight the transition probabilities in the rate matrix it based on the quality of each integer candidate, as evaluated by f(d1jt,) (G.)), steering the categorical distribution towards more promising integer assignments:
Een exp (J (diju. éi (G+) /v) Ran (d,", ; dfferr)
R(d.).))
Σ exp (f ((.:)) /bb)
where s is a temperature parameter. This updated rate matrix R is then used to sample the next integer state followed by the projection function for feasibility constraints, i.e., di+ < Project, , (d++
4.4 INTEGRATION WITH DOWNSTREAM SOLVERS
The output of the guided sampling process of FMIP is high-quality candidate solution (d1, c1) and a probability distribution over the integer variables. This provides a powerful warm-start for downstream solvers to efficiently search for better solutions. FMIP is fully compatible with wide range of downstream solvers and can provide powerful warm start for them to efficiently search for optimal solutions. For example, predict-and-Searee (Han et al. 2023)) employs the integer distribution to define promising search region and uses the continuous values as initial feasible point for the solver's ' LP relaxations.
5 EXPERIMENTS
5.1 EXPERIMENT SETTINGS
Datasets & Benchmarks We evaluate FMIP on comprehensive suite of eight MILP bench- marks. Five of them focus on classic combinatorial optimization problems: Combinatorial Auctions (CA) (Gasse et al. 2019), Generalized Independent Set (GIS) (Colombi et al. 2017),, Maximum Inde- pendent Set (MIS) (Gasse et al.12.011)) Fixed-Charge Multi- Commodity Network Flow (FCMNF) (He- witt et al. 2010), and Set Covering (SC) (Gasse al.) 2019). We also include two real-world MILP datasets with both binary and continuous variables from the NeurIPS ML4CO 2021 com- petition (Gasse et al. (0222): Load Balancing (LB) and Item Placement (IP). Finally, adopt the standard MIPLIB2017 benchmark (MIPLIB) which is one of the most common standards for MILP solver evaluation (Glexnnrr al. 2021a)
Baselines We compare FMIP against three representative learning-based methods and the commer- cial solver Gurobi (Gurobi Optimization, LLC, 2024). The learning-based baselines are:
SL: Standard supervised learning serves as a strong discriminative baseline. The model is trained directly predict the variable assignments in one-step manner.

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DIFUSCO: A sfate-of-the-ar diffusion model for generating solutions to integer linear pro- grams (Sun and Yang 2023: Feng et al.) 2024).
IP-Guided-Diff. A guided discrete diffusion framework designed for integer programs that incorporates problem-specifie guidance (Zeng et al. 2024).
Since DIFUSCO and P--Guide--Diff are designed for integer-only problems, we adapt them to the MILP setting by including continuous variables and their constraints in the graph structure for the backbone graph encoder. In this way, we allow the baselines to perceive both the continuous and integer variables for fair comparison.
FMIP and three baselines are all learning methods that are agnostic to backbone graph neural networks and downstream solvers. In later experiments, we employ four different backbone graph encoders, i.e. Tri-GCN (Gasse et al., 2019), Bi-GCN (Gasse et al.) 2019). GAT (Brody et al. 202 1T, and ClusterGCN (Chiang et al. 2019), with Tri-GCN as the default choice. Note that Tn- GCN and
bipartite topology; the former is distinguished by handling integer and continuous variables as heterogeneous node types with distinct parameters. As for downstream solvers, we adopt Neural Diving (ND) (Nair al. 2020), Predict-and-Search (PS) (Han et al. 2023
Bi-GCN both operate
PMVB (Chen et al. 2025) and Apollo-MILP (Apollo) [Lu et al. 2025).
2023 Liu et al. 2025 we report the objective value (OBJ) found by each method within fixed time limit. To compare performance, we first determine the best-known solution (BKS), defined as the best objective value found across all methods, including a long run of Gurobi. We then calculate the absolute primal gap: GAP (BJ- BKS). Since MIPLIB contains instances from diverse domains with very different optimal values, we report the relative primal gap for MIPLIB instead: Rel.GAP OBJ
Metrics Following previous works (Han et al.
1). A lower absolute/relative primal gap indicates better performance. Notably, we only report the Primal Gap instead of the Optimal Gap because FMIP acts as primal heuristic. Its objective is to quickly identify high-quality feasible solutions (improving the upper bound), rather than tightening the dual bound/optimality gap. This is the standard metric for evaluating heuristic improvements.
Implementation Details Due to the page limitation, we provide implementation details in Ap- pendix D including the backbone model architecture, training & inference hyperparameters of FMIP and baselines, as well as the downstream solver configuration.
5.2 OVERALL PERFORMANCE
We choose Tri-GCN as the backbone graph encoder and evaluate FMIP against baselines based on four downstream solvers (ND, PS. PMVB, and Apollo), with respective time limits of 400, 600, 600, and 800 seconds. The time limit for Gurobi is set to 3600 seconds. The results reported in T We observe that FMIP generally achieves the best performance among learning-based methods, substantially reducing the primal gap across all benchmarks and downstream solvers. On simpler benchmarks (i.e., CA, GIS) where multiple methods find the optimal solution, FMIP performs on par with the best baselines. However, on more challenging benchmarks (i.e., LB, IP), the benefits of our joint modeling and holistic guidance are clear, culminating in the significant performance gains in terms of both objective value and primal gap. Overall, our method brings a 41.34% improvement compared with the best machine learning baseline, calculated as the mean increase over all datasets and downstream solvers.
5.3 COMPATIBILITY ANALYSIS
The downstream solver compatibility of FMIP is already validated in Table I in Section 5.2, where FMIP consistently achieves the best performance across four different solvers. To further study the model compatibility of FMIP, we apply FMIP and baselines to four different backbone graph encoders on IP and LB benchmarks: Bi-GCN Gasse et al. 2019), Tri-GCN (Gasse et al. 2019). GAT (Brody ct al. 2021), and ClusterGCN (Chiang et al. 2019) We report the objective value (OBJ) metric in Table 2 where FMIP maintains its superior performance across all backbone graph encoders. This backbone independence demonstrates that FMIP acts a powerful and general framework that can enhance a wide range of GNN-based models for various downstream solvers, highlighting the fundamental value of its joint distribution modeling on both continuous and integer variables.
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Table The overall performance of FMIP and other baselines combined with different downstream solvers. Tri-GCN is chosen as the graph encoder. Best results are given in bold, and the second- best values are underlined. The relative improvement is computed Rel.Imprv. = (GAP best-bst GAPFMP)) (TAMP) -e- where GAPPmip is the gap of the FMIP method, and Ges is the gap of the best-performing baseline method. For MIPLIB, since we already give the relative primal gap (Rel.GAP) the relative improvement is computed as Rel.Imprv. = (Rel.GAP beat-bal ReI.GAP (MMP))
Downstream Solver
Training Method
CA
GIS
MIS
FCMNF
OBJ
GAP
OBJ
GAP
OBJ
GAP
OBJ
GAP
SL IP-Guide-DDiff
14691.07
1074.38
1250.10 10
573.60
440.30
9.70
1240317.40
160200.00
15132.65
632,80
1542.70
281.00
440.30
9,70
1226793.66 60
146676.25 20
ND(40X3)
DIFUSCO
15042.35
723.10
1487.50
336.20
439.20
10.80
1250323.CG 40
170206.00
FMIP (Ours)
15452.75
312.70
1554.20
269.50
449.60
0.40
1221710.50
141593.10
Rel.Imprv.
50.58%
4.09%
95.88%
3.47%
SL
15765.45
0.00
1783.00
40.70
450.00
0.00
1080117.40
0.00
IP Guided-Diff
15765.45
0,00
1815.30
8.40
449.60
0.40
1080117.44
0,00
PS(600s)
DIFUSCO
15761.85
3.60
1732.60
91.10
448.50
1.50
1120110.30
39992,90
Rmp. 15765.45
FMIP (Ouns)
0.00
1816.50
7.20
450.00
0.00
1080117.40
0.00)
0.00%
14.29%
0.00%
15765.45
1690.90
132.80
449,90
0.10
1080117.40
IP-Guided-Diff
15765.45
0.00
1695.90
127.50
448.30
1.70
1080117.40
0,00
PMVB(600S)
DIFUSCO
15745,25
20.20
1673.80
149.90
443.20
6.80
11 1114069.00
33951.60
FMIP (Ours)
15765.45
0.00
1695.90
127.80
450.00
0.00
1080117.40
0.00
Rel.Imprv
0.00%
0.00%
100.00%
0.00%
SL
15765.45
0.00
1823,70
0.00
448.20
1.80
1080117.40
0.00
JP-Guided-Diin
15765.45
1823.70
0.00
449.10
0.90
1080117.40
Apollo(800s)
DIFUSCO
15431.43
334.02
1763.24
60.46
447.50
2.50
1087517.20
7399.80
FMIP (Ours)
15765.45
0.00
1823.70
0.00
450.00
0.00
108011 17.40
0.00)
Rel.Imrr.
0.00%
0.00%
100.00%
0.00%
Gumbi(36000s
15765.45
0.00
1823.70 70
0.00
450.00
0.00
1080117.40
0.00
Downstream Solver
Training Method
SC
LB
IP
MIPLIB
OBJ
GAP
OBJ
GAP
OBJ 1
GAP 1
OBJ
ReL.GAP
SL 401.00
0.30
719.67
13.77
14.62
0.88
1619541 .12
6.21%
ND(400s)
IP-Guided-Diff
400.95
0,25
712.34
6.44
14.63
0.89
1609725.33
5.85%
DIFUSCO
402.10
1.40
717.73
11.83
15.41
1.67
1625188. 1225188.49
6.48%
FMIP (Ours)
400.80
0.10
706.30
0.40
13.99
0.25
1604917.80
5.42%
Rep.
60.00%
93.79%
71.59%
0.43%
SL.
400.80
0.10
749.60
43.70
15.34
1.60
1612105.7 77
5.67%
PS(600s)
Guided-Diif
400.75
0.05
725.34
19.44
15:21
1.47
1601226
5.31%
DIFUSCO
714.11
14.73
0.99
161
5.65%
FMIP (Ours)
400.70
0.00
705.90
0.00)
13.92
0.18
1595308.15
4.84%
Imprr
100,09
100.00%:
74.86%
0.47%
SL
422. 10
21.40
706.10
0.20
15.39
1.65
1620951 88
6.34%
PMVB(600s)
IP-Guided-Diff
405.50
4.80
706.30
0.40
15.17
1.43
1614667 00
5.88%
DIFUSCO
412.60
11.90
714.21
15.03
1.29
1628433.00
6. 6.62%
FMIP (Ours)
400.50
0.10
706.10
0.20
14.41
0.67
1608125.64
5.55%
Impry
97.92%
0.00%
48.06%
0.57%
SL
400.80
0,10
706.00
0.10
14.16
0.42
1604557. 34
4.98%
IP-Guided-Diff
405.50
4,80
705.95
0.05
14.33
0.59
1593821.50
4.55%
Apollo(800s)
DIFUSCO
403.30
709.32
3.42
14.01
0.27
1591488.19
4.43%
Impry 400.70
FMIP (Ours)
0.00
705.90
0.00
13.74
0.00
1586142.71
4.15%
100,00%
100.00%
100.00%
0.28%
Gurobi(3600s)
400.80
0.10
706.00
0.10
17.73
3,99
1522661.56
0.00%
5.4 INFERENCE EFFICIENCY
We report the averaged inference time of FMIP and baselines to generate a heuristic solution for one MILP problem different benchmarks. Tri-GCN is adopted as the default backbone graph encoder. Note that most downstream solvers (ND, PS, and PMVB) only need to obtain the heuristic solution based on inference process from the model. However, Apollo is special solver that requires the model to infer multiple times for each single MILP instance, with the problem size gradually reduced and the solution iteratively refined. In this experiment, we iterate times for Apollo.
As shown in Table 3 we report the inference time per MILP instance separately for ND/PS/PMVB and Apollo. Despite the superior performance of FMIP, its generative process is also highly efficient.
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Table The model compatibility study, where we apply FMIP and baselines to different backbone graph neural networks. We report the objective value (OBJ) metric. The best results are given in bold, and the second-best values are underlined.
Downstream
Training Method
Item Placement (IP)
Load Balancing (LB)
Solver
Bi-GCN
Tri-GCN
GAT
ChusteGCCN
Bi-GCN
Tri-GCN
GAT
ClusterGCN
14.62
14.65
719.67
719.20
720.15
IP-Grided-Dirt
14,82
14.63
14.63
14.63
713.32
712.34
712.20
713.10
ND(400s)
DIFUSCO
15.44
15.41
15.39
15.50
717.75
717.73
717.68
717.75
FMIP
14.11
13.99
13.95
14,06
707.24
706.30
706.50
706.50
SL
15.38
15.34
15.33
15.32
750.34
749.60
749.26
749.69
PS(600s)
I--Grided-Diff
15.24
15.21
15.21
15.21
726.04
725.34
725.46
725.72
DIFUSCO
14.73
14.73
14.76
14.81
714.01
714.11
714.77
714,90
FMIP
13.92
13.92
13.94
13.92
705.88
705.90
706.10
706.00
15.36
15.36
706.
706.30
706.30
PMVB(600s)
Di-i
15.25
15.17
15.15
15.17
706.62
706.30
706.20
706.30
DIFUSCO
15.07
15.03
15.03
15.09
714.89
714.21
714.01
714.55
FMIP
14.45
14.41
14.42
14.39
706.32
706.10
706.10
706.20
SL
14.19
14.16
14.11
14.19
706.30
706.25
706.20
706.30
Apollo(800s)
IF-Guided-Diff
14.36
14.03
14.33
14,40
706.00
205.95
706.10
706.
DIFUSCO
14.01
14.01
13.99
14.05
709. 92
709.32
709.40
709.40
FMIP
13.72
13.74
13.74
13.75
705.90
705.90
705.90
706.00
Table 3: The inference time (s) per MILP instance of FMIP and baselines. We only compare the neural
model inference here, including the computation of guidance, and exclude the time that downstream solvers take.
Downstream Solver
Training Method
CA
GIS
MIS
FCMNF
SC
LB
IP
MIPLIB
ND(400s), PS(600s), PMVB(6600s
SL DIFUSCO
0.047
0.089
0.138
0,065
0.087
0.088
0.023
0.115
0.187
0.652
0.340
0.238
0.412
1.154
0.093
1.561
D--uuidd-Diiff
0.213
0.712
0.452
0.374
1.781
0.117
FMIP
0.281
0.761
0.476
0,310
0.623
1.294
0.176
2.043
Apollo(800s)
SL
0.082
0.273
0.184
0,310
0.209
0.213
0.093
0.327
DIFUSCO
0.512
2.125
1.341
0.545
1.112
3.129
0.153
3.231
P-Guuded-Difff
0.612
2.173
1.549
0.634
1.268
3.151
0.167
FMIP
0.718
2.060
1.805
0.546
1.782
3.229
0.121
4.812
The inference time of FMIP is comparable to that of other generative baselines (i.e., DIFUSCO and IP-Guided-Diff), Crucially, this time represents negligible fraction (often less than 1%) of the total time spent by the downstream solver, which can be hundreds thousands of seconds. FMIP thus provides its substantial performance improvements with minimal computational overhead, striking an excellent balance between solution quality and efficiency.
5.5 IN-DEPTH ANALYSIS
Ablation Study. We derive four variants our proposed FMIP: (i) removing the objective guidance in Eq. 3 (w/o Feasibility Guidance), (ii) removing the objective guidance in Eq. (w/o Objective Guidance), (iii) removing the entire holistic guidance mechanism (w// Guidance), and (iv) reverting integer-only model by disabling continuous variable generation (w/o Continuous). which inherently prevents the application of holistic guidance. As shown in Table 4 the full version of FMIP significantly outperforms all ablated variants, confirming that both the joint generation of all variables and the holistic guidance mechanism are critical to the performance gain. Notably, the "w/o Continuous" variant suffers from the worst performance, providing direct evidence that capturing the coupled relationship between integer and continuous variables is essential for high-quality solution prediction MILP.
Impact of Sampling Steps. We analyze the effect of the number of sampling steps on the heuristic solution quality. Specifically, we feed the heuristic solutions from different sampling steps into the downstream solvers, and report the primal gap metric (i.e., GAP) in Figure We can observe that, at the beginning, more sampling steps lead better solutions as the model has more opportunities for refinement. However, the quality of heuristic solution starts to worsen when plethora of steps are performed. We attribute this phenomenon to the trade-off between generative refinement and search diversity. An excessive number of sampling steps might cause the guidance make the predicted variable distribution overly sharp, which reduces the diversity the solutions being explored and prematurely trap the search in local optimum. Alternatively, this degradation may stem from

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Published as a conference paper at ICLR 2026
Table 4: Ablation study of FMIP and its variants. The best results are given in bold, and the second- best values are underlined.
ND
PS
PMVB
Apollo
Variant
OBJ J
GAP
OBJ
GAP
OBJ 1
GAP 1
OBJ 1
GAP
Full FMIP
13,99
0.25
13.92
0.18
14.41
0.67
13.74
0.00
w/o Feasibility Guidance
14.75
1.01
14.67
0.93
15.47
1.73
14.97
1.23
Objective Guidance
14.76
8.58
w/o Guidance
14.58
0.84
13.98
0.24
14.45
0.71
14.11
0.37
w/o Continuous
14.43
0.69
14.28
0.54
14.95
1.21
14.45
0.71
the balance between objective value and feasibility in the target function 5 which is discussed in the subsequent part.
Performance on SC Datanet
Performance on NIS Dataset
Performance on 1B Dativet
Performaane on Datiowt
1.55
2 2ots
0.0
Sampleg Step
Suaggung Skop
Sumplig Slop
Sangling Slop
Aprila
Figure 3: The performance of FMIP with downstream solvers w.r.t. different sampling steps. We report the absolute primal gap (GAP) as the metric.
Adaptive Guidance Balancing. We investigate the potential imbalance between objective and feasibility guidance terms during iterative generation. We hypothesize that a fixed coefficient ? (defaulting to 50) leads to performance degradation over longer sampling trajectories. To address this, we introduce an adaptive balancing strategy that dynamically adjusts y at each generation step. Specifically, we compare the magnitude of the objective guidance term with that of the constraint- violation term: if the objective term dominates, we scale up y by a factor a (i.e., 7 f- Y X a) to enforce stronger constraint satisfaction; conversely, if the constraint term prevails, we scale down (i.e.. 7 + 7/a). This adjustment is cumulative across sampling steps, allowing the model to dynamically recalibrate the guidance focus between optimality and feasibility. We evaluate this strategy on the Item Placement dataset using the PS solver, varying a and the number of steps. As shown in 5 with fixed y, the objective value worsens significantly as steps increase (e.g., from 11.79 at Step- 12 to 13.25 at Step-32). However, the adaptive strategy (a > L) effectively mitigates this degradation, maintaining competitive performance even deeper generation steps (e.g.. 11.92 at Step-32 with a = 2.0). These results validate that dynamically balancing guidance is crucial for stabilizing long- traarrajectory generation.
Table 5: Impact of Adaptive Balancing Strategy on Objective Value (Lower Better) across different sampling steps on the IP dataset.
Strategy
Step-12
Step-24
Step-32
Fixed y
11.79
12.39
13.25
Adaptive (a = 1.1)
11.93
12.35
12.05
Adaptive (a = =1.5)
11.99
11.85
12.68
Adaptive (a = 2.0)
12.19
11.95
11.92
CONCLUSION
In this paper, we propose Joint Continuous-Integer Flow for Mixed Integer Linear Programming (i.e., FMIP), which the first generative framework to model the joint distribution of both integer and continuous variables. Based on this, we further design the holistic guidance mechanism that uses instance-specific objective and constraint information to steer the generative process toward higher- quality heuristic solutions. As powerful generative learning framework, FMIP is fully compatible with arbitrary backbone graph encoders and various downstream solvers, demonstrating its great potential for real-world MILP applications. Extensive experiments show that FMIP sets new state- of-the-art performance for learning- based MILP heuristics, reducing the primal gap by 41.34% on average compared to the best baseline method. Two key directions for future work are: enhancing the graph representation to better capture task-specific features, and developing a customized solver tailored our FMIP framework to optimize solution quality and computational efficiency.
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ACKNOWLEDGMENT
Lin's research is partially supported by the National Natural Science Foundation of China (NSFC) [Grant NSFC-624B2096, 72542012, 72595872]. Ge's research is partially supported by the National Natural Science Foundation of China (NSFC) [Grant NSFCC72225009, 72394360, 72394365).
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A LLM USAGE DECLARATION
In this submission, we used a Large Language Model (LLM) solely for language refinement and text polishing. The LLM was employed to enhance the clarity, flow, and readability of the manuscript, it did not contribute to the ideation, analysis, or generation of scientific content. All ideas, interpretations, and results presented in the paper are solely the work of the authors. No content generated by the LLM was used to fabricate facts or contribute to the research findings.
B GRAPH REPRESENTATION DETAILS
Building previous work, we select similar graph structure. To more naturally accommodates joint generation purpose, we explicitly distinguishing three types nodes: Integer Variables (Ivar), Continuous Variables (Cvar), and Constraints (con). Edges, denoted Ctj, are drawn between the variable nodes (both Ivar and Cvar) and the constraint nodes to capture their algebraic relationships within the MILP formulation. The edge weights eij correspond to the coefficients of variables in their associated constraints. The features of nodes and edges of a MILP graph G (Vivar: Vevar Veon E)
are:
Ivar/Cvar Nodes (Viar\ Vear): The feature vector for variable Uy E Vivar U Vevar is defined as:
ho;
W0ss
hlb;
hub;
E TRS
objective coefficiint
upper bound
1--o
uu <
where w, L, defined in Eq.
Constraint Nodes (Vem)): :: The feature for constraint con, is scalar by E R. representing its right- hand side constant.
Edges (E): We construct sparse bipartite graph based the nonzero entries of the constraint
matrix A. Specifically, an edge is added between variable node j and constraint node if and only if A H.. The edge feature is set as the corresponding coefficient, i.e., the edge from j to carries weight ji.
When processing with graph neural network, the state in generative model æ; = (de, Ct) is incorporated into the features of the corresponding variable nodes in G. Specifically, for each variable node U, E Vvar Vear. its static feature vector h,, from the MILP graph is augmented with its current value from the solution vector forming the time-dependent feature vector h [h. _z. R This implies that the static variable features h, are 5-dimensional. Constraint node features are typically static. Figure 4 provides simple example illustrating the structure of solution graph derived from an MILP instance.
min
4r F2 3r, + x2 I 1,
hep = (4, (0,,0,,,,,01,,x,
s.t.
lnteggr Variables
21 +2
Reon, (1)
heon, = (2)
2 $155, 5,
Constraints
0 82 3,
Comtinuuus Vartables
21 E Z.
hpge = (1,0,3,1,1, (2)
(a) Problem
(b) Graph representation
Figure 4: Toy example of a MILP instance and its graph representation in Flow Matching.
C GRAPH NEURAL NETWORKS
The input to our neural network is a tuple (G., t), where G, represents the solution graph as described in Section B This graph encodes the relationships between variables and constraints, effectively capturing the structure of the MILP problem. The scalars are temporal parameters associated with the integer and continuous variables, respectively, with both t O0, 1]. These time parameters introduced to enable a dynamic, progressive refinement the solution space during the training process, aligning with the flow matching paradigm where time plays critical role in guiding the
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evolution of the generated solution. We'll details our neural networks structure in the following subsection.
C.I GCNCONV LAYER
We adopt the following mathematical notations for the following parts: Linear. denotes learnable affine transformation (linear layer), LN.() represents layer normalization, and GELU(.) indicates Gaussian Error Linear Unit activation. The element-wise sigmoid function is written as (-),, while @ signifies element-wise multiplication. Vector concatenation is denoted by [+; :I. and Tes E {0, serves indicator variable for residual connection usage.
The output feature Hi of a target node v computed as:
hy LNout (Linear (GELU (Linearo1 (L @ a1 (1 9v) O a2) hol)))) + Lres her where the aggregated messages from source node types & 1, are defined as:
0Ak =
Σ
Linearinal
GELU [LNNN (Linear, (h.,) + Linears, (h + Linear (e (k)
(ENa(e)
and the gating vector is computed by: 9. = a [Linearsy[lan)) a2l)).
C.2 MODEL STRUCTURE
Overview We propose graph-based neural architecture tailored for mixed-variable optimization problems. The model captures the interactions among integer variables, continuous variables, and constraints using iterative message passing with temporal conditioning.
Initial Embeddings
We define three types of nodes:
Integer variables (Viva): Alvar nodes, with features X RR Fllvrr
Continuous variables (Vevr): : necar nodes, with features X E IR Xdcva
Constraints (C.on): Thcon nodes, with features X IR Reze Xdan
Initial node embeddings are computed via type-specific MLPs:
(0) MLPivar (X Iva)) h MLP (Xevr), h (0) MLP MLPcoo (Xcon); Ivar
con
where each embedding lies in R.xh with denoting the shared hidden dimension.
The time embedding e; E Ih is initialized using positional encoding Vaswani et al. 2017). This embedding broadcasted and added to the variable node features at every iteration.
Message Passing Layers For each layer é é 1,..., L. we update the node representations via message passing scheme over bipartite graph with three node types. The relevant edge sets are defined as:
Eivar2con S Vivar X Ceon integer-to-cconstriie edges
Ecvar2eon C VCvar X Cron: continuuuss-to-connstiiiiii edges
Reverse edges Econ2Ivar; EcantCvar constraint-oo-variaaae edges for backward message flow
The node updates at layer l are:
h = MLP! (ct).
h (E) (f-1) + h (2) MLP(() (TriConv (bfer") (6-1) (e-11) (f-1)) E Ecvar2eom) ));
h. (8) h ((-1) h MLP(() (BiCony, (h( h! (e-1) E)va22)); [Ivar, Cvar).
TriConv message aggregation module that aggregates signals from two types of source nodes into target nodes. Bi BiConv has the same architecture as TriConv, but aggregates signals from one type of source nodes into target nodes.
Output Layer After L layers of message passing, we apply type-specific MLP heads to predict outputs from the final node embeddings:
Ovaa
MLPivar
out Iaar
OCvar
MLPCCar Cvar out
Cvar)
where Ojarr E R Thu x dse and Ocvar E Pea x x are the prediction outputs for integer and continuous variables, respectively.
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D IMPLEMENTATION DETAILS
We'll 'l present the implementation details in this section, which contains infrastructure and hyperpa- rameter in our proposed methods.
D.1 TRAINING DETAILS
Training labels for FMIP and other baselines were generated by solving the MILP instances using COPT (Ge et al. 2024), an outprrformig MILP solver (Gleixner et al. 2021b), with a time limit of 3600 seconds per instance and 12 threads. The training loss for the GNN baseline is binary cross entropy for the prediction of binary variables, and the training loss for FMIP is described in Eq 4 For MIPLIB, the training set is collected from all other benchmarks due to its diverse problem types. For the other datasets, the training set is split from the entire data with 9:1 ratio.
D.2 INFRASTRUCTURE
Our flow model is implemented PyTorch TarchePasskk et al. 2019) and PyTorch Geometrie(Fey and Lenssen 2019) and trained on single NVIDIA H100 GPU.For CPU, we we use 12 cores of an Intel Xeon Platinum 8469C at 2.60 GHz CPU with 512 GB RAM. We select Gurobi(Gurobi Optimization LLC 2024) which is the most famous and well well-implement MILP solver, in Decoding Strategy and baseline methods due to its high efficiency. For all experiments, we set 12 threads for backend solver, Gurobi, which is standard setting (Gleixner al. 2021b).
Table Experimental Parameters Comparison
Method
ND
PS
PMVB
Apollo
(Parameters)
[ a]
[ko, k1, A]
88 r]
[kg, k1, A, K]
Cauctions
50, 0.1
003, 0.06, 0.3]
00.7,0.9
0.3, 0.06, 0.3,
GISP
50, 0.1
0..2, 0.2
00.7,0.9
0.2,, 0.02, 0.2,
Independent Set
50, 0.1
[0.3, 0.3]
10.7.0.9
10.3, 0.3,
FCMNF
50, 0.1
00.3,0.03. 0.2
[0.7,0.9
00.3, 0.03, 0.2,
Item Placement
50, 0.1
0.3,
00.7.0.9
00.3, 0.08, 0.4,2
Load Balancing
50, 0.1
0.2,
0.7,
[0.2, 0..,,.02,
Set Covering
50, 0.1
0.3,0.04,0.2
[0.7,0.9]
0033, 0.04, 0.2,
D.3 DOWNSTREAM SOLVERS
Table 6 summarizes the hyperparameter settings used by each downstream solver across different problem instances. The meaning of each parameter (e.g., K, a, kg. A, etc.) is consistent with the notation defined in Appendix [E] where detailed deseriptions of downstream solvers and their hyperparameters are provided. For a fair comparison, we use the same parameters for the downstream solvers when evaluating FMIP and all baselines.
D.4 FMIP
This section outlines the key hyperparame- ters employed during model training and in- ference, covering: (1) general training config- urations (e.g., learning rate, batch size), (2) neural network architecture specifications, and (3) inference-specifie settings such as sampling strategies and guidance control. The fixed de- fault parameters are provided in Table while two critical parameters, Batch Size and Guid- ance Temperature, are dynamically adjusted based dataset characteristics, detailed in fol- lowing parts.
Table 7: Training and Inference Configuration for FMIP
Parameter Description
Value
Training epochs Learning rate Weight decay Learning rate scheduler
300
2e-4
1e-4
cosine-decay
GNN layers Hidden dimension
12
64
Inference schedule
cosine
Inference Steps
12
Choice of Batch Size In practice, Batch size dynamically adjusted according to the prob- lem scale and available GPU memory. We adopt the largest feasible batch size that fits into mem- ory maximize hardware utilization and training efficiency.
Integer guidance temperature (x)
0.1, 1, 10
Continuous guidance stepsize (p)
0.1
Weight of constraint violation (n) 50
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Guidance Temperature We temperature coeffient (τ) is introduced control the strength of guidance during inference, where lower temperature represents higher strength. This parameter is selected based on validation performance. Specifically, we set the temperature to 0.1 for the problem Load Balancing and Item Placement, 10 for the Set Covering problem, and for all other problem types.
E DOWNSTREAM SOLVER ALGORITHM
In this section, we introduction detailed algorithms employed in our experiments, including Neural Diving, Predict-and-Seaut PMVB and Apollo-MILP.
E.1 NEURAL DIVING
For a new instance, the neural network first generates multiple partial variable assignments based on predicted probability distributions. Let V denote the of variables. The algorithm fixes subset S C V where:
S a [VI (a E (0,
and generates K parallel sub-MIPs through {S:} K assignments. Through a selective prediction mechanism, it decides which variables fix and samples their specific values, forming various assignment combinations that only involve subsets of variables. Subsequently, each partial assignment transformed into a smaller sub-MII problem by fixing these assigned variables. These sub-MII problems are then solved in parallel using traditional solvers to generate multiple complete feasible solutions. Finally, the solution with the optimal objective function value selected as the final result. This process combines the predictive capabilities of neural networks with the efficiency of traditional solvers, achieving an end-to-end workflow from partial assignments to optimal solutions.
E.2 PREDICT-AND-SEARCH
In the predict phase, graph neural network estimates binary variable assignments â, E {0, 1}. During the search phase, the algorithm constructs trust region based on the predictions, restricting the original problem to feasible solution space within the neighborhood of the predicted solutions. The trust region is defined as follows:
To { ig S ko}
(variables near 0)
Ti {x, 22 - k1} (variables near I)
Σ I(aj 1) + Σ - < A (1T6l + (Til)
JET
JET:
where ko: k (0, 1) control selection thresholds and defines permissible deviation. Specifically, by adding constraints (e.g., limiting the number of variables with scores close to 0 or 1 and controlling deviations from predicted values), the problem is transformed into smaller subproblem, which is then solved using traditional solvers to efficiently search for high-quality feasible solutions. Compared to methods that directly fix variables, the trust region strategy retains flexibility to avoid infeasibility while improving solution quality through localized search.
E.3 PMVB
The PMVB (Probabilistic Multi-Vante Variable Cardinality Branching) method leverages predictions from graph neural network to construct statistically grounded branching constraints. Specifically, the binary variables are partitioned into two disjoint sets based on their predicted probabilities:
u= {j 9, 2 T), L {j 0J 1--r]
where 7 E (0.5, 1] is confidence threshold indicating how certain the model is about a variable being or 0, respectively. Using the principles of risk pooling and concentration inequalities, the algorithm constructs two soft constraints that serve as branching hyperplanes:
Cu
Σ U1 2
(1
8) Σ EbI -
Cc
ME JEC
6
> jEC
B
+
JEM
jEur
where r
[S] (2(688) for (4, L). and is a tunable confidence parameter.
2
These hyperplanes probabilistically constrain the variable assignments, forming a high-confidence trust region that filters out unlikely configurations while preserving feasible and high-quality candi- dates. By intersecting the feasible region with both C, and Cg. the algorithm defines subproblem Poy.Ce that is both reduced in size and refined in quality.
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This subproblem is then passed to traditional MILP solver for efficient resolution. Compared to methods that directly fix variables, PMVB offers greater robustness by retaining feasible flexibility while introducing statistically grounded directional guidance. As result, it effectively balances confidence-drived pruning with solution diversity, improving both computational efficiency and solution quality in challenging problem instances.
E,4 APOLLO-MILP
The algorithm begins by employing a graph neural network to predict values for currently unfixed variables, yielding predicted solution. Subsequently, the Predict-and-Sed and-rercc-h method is applied to solve subproblems derived from this prediction, generating reference solution. By comparing the predicted and reference solutions, variables with identical values in both solutions are fixed, thereby constructing a reduced problem. At each iteration E {I,..., K the algorithm applies parameters [K("), ( A [0] in Predict-and-Search algorithm, where K controls total iterations. This process iteratively repeats, progressively identifying high-quality partial solutions and expanding the subset of fixed variables reduce problem dimensionality.
Throughout the iterations, Apollo-MILP alternates between prediction and correction steps, con- tinuously refining the predicted solution and reducing the complexity of the MILP problem. This approach ensures optimality while significantly enhancing both solution quality and computational efficiency.
F PSEUDO CODE FOR FMIP INFERENCE
Algorithm 1 Inference Phase of FMIP
Require:
Rectified flow model Go
Target function f. guidance strength temperature T
Number of guidance steps Nier
Number of inference steps Nin. temporal step size = procedure MAIN
= [A/ =0, At2
Aln.
2: Convert MILP instance to Graph G
3: Get I, u E Rn for constraints S x S u. and let represent [4, u|
4: Sample do ~ NO, I[])) Co ~ Uniform([1,2,... K})12
5: (dg, ca] <- Project& ((ng co))
6:
Go < (G, do, co)
7:
for dt E [At1, At2, AtN,] do
Simulate Fokker-Planck & Kolmogorov Equations
8:
p(d1))) Cije (Go)
9:
Integer Variable guidance
10:
Sample {di d1jr. for i= 1 do i(d)),))
d d() p(d1j()
11:
DR f(d (dd Calel (1(1) (1
12:
E M(dieerss))))
Sample d:e Cat (8(d()) (d(d)4rr d(") R((d)) di At) end for
13:
14:
15:
Continuous Variable guidance
(k) Sample d1jt Cat [f((d(()) 1k=1 c111)
16:
17:
for j = to Niter do
18:
C: Projects (c+ + pVaf (dijer go (dl, cile))
19:
end for
20:
Cije +- go (d,, Ct)c
21:
(,(c,) + (c Enplen G.l) = Cij--c
I-t
22:
c+tt + Project (c: + (((cs) At)
23:
t At
24:
end for
25:
Output: X1, a1
26:
end procedure
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G ADDITIONAL EXPERIMENTS
G.1 BREAKDOWN ANALYSIS ON MIPLIB 2017 BENCHMARK
To further investigate scalability and analyze failure cases, partitioned the MIPLIB 2017 test into five subsets based on the scale of the MILP instances. We report the relative primal gap. Rel.GAP, of FMIP these subsets w.r.t the Best Known Solution (BKS) provided by MIPLIB official website.
Table 8: Relative Primal Gap of FMIP subsets of varying instance sizes.
Instance Size
FMIP Rel. GAP
Best Baseline Rel.
50 3,000
4.70%
5.03%
3,000 8,500
4.81 %
5.19%
8,500 20,000
4.86%
5,445%
20,000 70,000
9.04%
9.10%
70,000 300,000
23.26%
25.38%
The results in Table reveal distinct performance dichotomy: while FMIP maintains robust stability (Rel. GAP < 5%) across small to medium-large instances (up to 20,000 variables), performance degrades ultra-large-scale instances (> 70,000 variables). In these extreme regimes, the GAP widens to ~ 23% as the exponentially increasing complexity of the constraint landscape challenges the model's generative precision.
Despite the degradation extreme cases, FMIP still serves as the SOTA primal heuristic predictor to provide feasible starting points, making it significantly faster than cold-start solver. This analysis provides a clear direction for future work, i.e., improving backbone scalability for ultra-large instances.
G.2 SCALABILITY ON EXTREMELY HARD INSTANCES
We further test FMIP combined with Predict-and-Seee large-scale Set Covering (SC) problems (up to 1,000,000 variables), which are significantly larger and harder than the training set (around 5,000 variables). The time limit of FMIP is set 3600s, which is empirically enough for superior performance in such extreme cases. The time limit of Gurobi is set according the problem scale.
(FMIP-PS) against Gurobi extremely
Table 9: Objective Value Time w.r.t. Different Test Problem Scale (Lower is Better).
(a) #Constraints 50,000; #Variables 500,000
Solver
60s
600s
1800s
3600s
Gurobi
785,972
785,972
53,008
52,877
FMIP-PS
1,335,583
52,984
52,905
52,859
(b) #Constraints 100,000; #Variables 500,000
Solver
60s
1800s
3600s
7200s
Gurobi
1,015,955
1,015,955 1,007,549
1,015,955 76,579
1,015,955
FMIP-PS
1,007,549
-Stop-
(c) #Constraints = 500,000; #Variables = 500,000
Solver
60s
1800s
3600s
7200s
25000s
Gurobi
1,654,768
818,547
469,130
469,130
469,130
FMIP-PS
1,007,549
184,821
184,821
Stop
Stop
(d) #Constraints 100,000; #Variables
1,000,000
Solver
60s
1800s
3600s
7200s
25000s
Gurobi
1,022,627
1,022,627
1,022,627 43,113
1,022,627
44,039
FMIP-PS
192,417
192,417
-Stop
Stop
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We can observe that, for these hard instances, Gurobi often stagnates converges very slowly. FMIP could help Gurobi to find solutions that are orders of magnitude better (e.g., ~ 43k vs 1M) or reaches comparable solutions hours faster. These results further demonstrate the superiority of our proposed FMIP to further boost Gurobi, and validate the core practice of accelerating downstream solvers with Al powers.
H RELATIONSHIPS BETWEEN FMIP AND EXACT SOLVERS LIKE GUROBI
We would like to further clarify that FMIP is designed to enhance exact downstream solvers like Gurobi, instead of replacing them.
The exact solver, Gurobi, is the backend engine in our experiments. The "Downstream Solvers" in our paper (i.e., Neural Diving, Sedict-and-Seardee Apollo) are, in fact, neural decomposition frameworks that wrap the exact solver Gurobi as improved version. They do not replace Gurobi; rather, they use the heuristic solution predicted by FMIP decompose the original hard MILP into smaller sub-MIPs, which are then solved by Gurobi. Therefore, our method is designed precisely to function with wrapped or exact solvers, enhancing their ability to find high-quality feasible solutions in restricted search spaces.
21


Reviews:
[{"review_id": "COoRqFlZkX", "reviewer": "Reviewer_3TSm", "summary": "The paper introduces FMIP (Joint Continuous-Integer Flow for Mixed-Integer Linear Programming), a generative framework that addresses limitations in existing MILP heuristics by modeling the joint distribution of both integer and continuous variables using conditional flow matching. This approach captures the interdependencies between variable types, enabling a holistic guidance mechanism during inference that steers solutions toward optimality and feasibility via objective function and constraint feedback. FMIP, integrated with various graph neural network backbones and downstream solvers, is evaluated on some benchmarks.", "questions": "None", "limitations": "", "rating": 6, "confidence": 2, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "The claimed empirical performance is good, achieving decent relative improvement in primal gap over state-of-the-art baselines on some benchmarks. The framework's compatibility with multiple GNN architectures and solvers, as shown in ablation studies and compatibility analyses, highlights its flexibility and generalizability. Additionally, the holistic guidance, supported by gradient-based updates for continuous variables and sampling-reweighting for integers, effectively leverages complete solution candidates at each step, leading to higher-quality heuristics.", "weaknesses": "FMIP's focus on bounded integer variables limits its applicability to general MILP problems with unbounded or non-binary integers. The reliance on flow matching introduces higher inference times compared to discriminative baselines like supervised learning. While ablations confirm the value of joint modeling and guidance, the paper could benefit from deeper analysis on failure cases or scalability to very large-scale instances beyond the tested benchmarks.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "0YNTHvJbOj", "reviewer": "Reviewer_1do2", "summary": "This manuscript is an incremental extension of [1], with the following core enhancements: i) an extention from integer linear programs (ILP) to mixed-integer linear programs (MILP), and ii) replacing diffusion models with flow matching models. The authors conduct extensive experiments, demonstrating improvements over existing methods.\n\n\n[1] Zeng, Hao, et al. \"Effective generation of feasible solutions for integer programming via guided diffusion.\" ACM SIGKDD 2024.", "questions": "1. Does the inference time in Table 3 include the time cost of the guidance process?\n2. In Table 4, how does the \"w/o Continuous\" variant perform the guidance? Without continuous variables, how are constraint violations and objective value calculated?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 4, "presentation": 4, "contribution": 2, "strengths": "1. The paper is well-organized with a clear logical flow.\n2. Extending the application of generative methods from ILP to MILP is a natural progression.\n3. Empirical experiments are comprehensive, and the results show improvements over the selected baselines.", "weaknesses": "I have reviewed this paper at NeurIPS 2025, and I am satisfied with the revisions made by the authors. I do not have any major concerns, but I have a minor suggestion:\n\nThe paper states that \"existing generative methods for MILP suffer from a critical limitation: they model the distribution of only the integer variables.\" However, the transition from ILP to MILP seems relatively straightforward, and this limitation may not constitute a major challenge. I recommend that the authors emphasize the more difficult aspects and make the challenges clearer to the reader.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "S2wzsX0TAT", "reviewer": "Reviewer_n8PZ", "summary": "This paper introduces FMIP a flow model for MILP solving, cast at sampling from a generative model, that defines a method to sample from joint distributions over discrete and continuous variable assignments. This can help capturing interactions between these two sets of variables, which is often missing in previous works on generative models for MILPs.\nThe model is based on a 'tripartite-graph'  a variation of the standard bipartite graph where integer and continuous variables are mapped to two disjoint set of nodes.\n\nHowever, while the distribution is joint, it is assumed to be factorized and amounts to one independent prediction per variable (discrete or continuous). Only the conditioning is from a complete, and noisy, assignment.\n\nAdditionally, authors introduce guidance mechanisms for the two types of\nvariables helping steer the model's trajectories towards feasible assignments.\nFor the continuous variables the guidance is based on projected gradient descent to enforce box constraints (the other constraints are taken into account via the violation penalty)\nFor integer variables, authors use a sample-and-reweight method where the denoiser is used to generate several candidates which are evaluated. The rates are updated to promote (ie, increase the probability) of candidates which combine good objective value and low constraint violation. The final samples are also projected to enforce box constraints.\n\n\n\nEventually, FMIP returns an assignment for the continuous relaxation of the\ninput MILP, and must be processed further by a solver. In that sense, FMIP\nremains a tool to warm-start a solver.\n\n\nTypos:\n- The two first references to Gasse et al. are the same\n- The two first references to Taoan Huang et al. are the same\n- l. 712 missing space before Edges\n- l. 728 why $> 0$ and not $\\neq 0$ ?\n- l. 753 $t$ and $t$ ??", "questions": "1. Where is the 41.34% improvement mentioned in the abstract and the conclusion found in the experiments? Maybe it's lost \n2. In Table 4, how is the system \"w/o Continuous\" implemented? How does it handle continuous variables? This is important since, as mentioned by authors, this performance gap gives a direct evidence that modelling joint distributions as advocated in FMIP is important.\n3. Why do you call your graph a tripartite graph instead of a bipartite graph with continuous/discrete binary features for variable nodes? It overcomplicates the model and makes the reader thinks edges between discrete and continuous nodes are allowed.\n   Besides, line 726 mentions a bipartite graph\n4.The discussion lines 468-474 is interesting. Could an additional reason to explain why the number of sampling steps may have a detrimental effect is that the coefficient $\\gamma$ given to the violation (and possibly the measure of the violation itself based on the max operator) may lead to a situation where the function $f$ favors constraints over objective?", "limitations": "", "rating": 4, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "This paper identifies and  highlights one of the blind posts from previous generative models for MILPs: the lack of a proper treatment  of continuous variables. I believe this is the main contribution of the paper", "weaknesses": "- The model is straightforward and amounts to the concatenation of a continuous flow and a discrete flows that can both be conditioned on predicted assignments.\n- The tri-partite representation is surprising and questionable (see question below)", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "xOvGmmJ1or", "reviewer": "Reviewer_t9Xo", "summary": "FMIP uses a multimodal flow matching generative model to learn the distribution of high-quality MILP solutions in a combined continuous–discrete space. The author proposes a **“holistic” guidance mechanism** that steers the generation process using the MILP’s objective function and constraint satisfaction feedback, refining candidate solutions toward feasibility and optimality during sampling. This approach is expected to addresses the limitation of previous graph neural network (GNN) predictors that predicted only integer variables and left continuous ones to a solver, thereby missing the intricate coupling between them .", "questions": "1. The guidance combines the objective with squared constraint violations via a hyperparameter $\\gamma$. Please clarify how it balances objective vs. violation terms, its sensitivity, and the tuning/selection procedure used.\n2. The reweighting step appears to require evaluating continuous variables, which is extra computational efforts. If many continuous variables are functionally dependent or cheaply recoverable, why is explicit prediction necessary? Could a GNN/GAT that aggregates global information suffice without predicting continuous components? Is it necessary because of the generative model? Please clarify the additional computational cost or justify the benefit of jointly predicting discrete and continuous variables (e.g., tighter guidance, reduced repair time, better warm starts).", "limitations": "", "rating": 4, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "1. The joint modeling of integer variables and continuous variables is a notable contribution, where previous works usually only focus on discrete value predictions, which ignores some global relationships.\n2. The author proposed the holistic guidance mechanism integrated into FMIP’s sampling. During inference, FMIP uses the MILP’s objective and constraint violations to guide the generative trajectory, which is expected to produce better predictions.", "weaknesses": "1. Given the context where integer–continuous coupling is important, the motivation for a generative approach is under-developed. Why is a generative model preferable to discriminative predictors under identical solver budgets? Why is the Flow Matching used in the proposed framework?\n2. In Tables 1–2 the downstream solver time limits should be standardized (e.g., a flat 1000 s for all methods) to ensure fairness; the mixed 400/600 s budgets are short for some instances and confound comparisons. A unified budget would make the FMIP gains more persuasive.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "RcGRF8QoD4", "reviewer": "Reviewer_rMCB", "summary": "The authors study mixed-integer optimization problems (MILPS) which are computationally hard to solve to optimality in general. Solvers for MILPs benefit from good heuristic start solutions, which can be generated by using state-of-the-art learning methods. The authors propose a generative model which is able to generate good feasible solutions by predicting (compared to previous models) both, integer and continuous variables. The performance of the model is tested on a wide variety of instances and compared to several benchmarks.", "questions": "- Why don't you use any exact downstream solvers as Gurobi, SCIP etc.?\n- Looking at the experiments, why should I use your (very complex) model involving data collection and training instead of just running Gurobi?\n- How good is your model for hard instances which are harder than the instances in the training set?\n- How much does your model improve the Gurobi process compared to the standard heuristics implemented in Gurobi?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "The paper is very clearly written and all concepts are presented in a detailed but concise way. The methodology is very sophisticated and the idea of incorporating the continuous variables into the generative model is interesting and hence the whole paper provides an important contribution to the knowledge of the field. The experiments are extensive, although there are some subtle issues which should be improved.", "weaknesses": "1. The goal of the paper is to provide (exact) solvers with good feasible solutions. Hence I don't understand why there is no state-of-the-art exact MILP solver used as downstream solver (e.g. Gurobi, SCIP or HiGHS)? It would be important to see on very hard instances, how much the predicted solutions improve the performance of the exact solver (e.g. in terms of optimality gap after 1 hour). Solvers as Gurobi have sophisticated methods implemented to search for good feasible solutions. If your extremely complex model (which has to be trained etc.) cannot provide better solutions then it is useless for exact solvers.\n\n2. In the experiments Gurobi is able to provide the same objective values as your model for nearly all instances (except LB and IP, although on LB it nearly achieves the same value). This indicates that the instances which are solved here are quite easy for exact solvers as Gurobi and at the same time Gurobi provides you with an optimality gap (which you should report). So looking at the results as a user I would prefer using Gurobi, which maybe has to run for 3600 seconds but on the other hand it is extremely simple to use: there is no training involved and it provides an optimality gap. As mentioned above, it would be interesting to see how good your predicted solutions are for harder instances.\n\n3. It is not clear from the main paper, what data the model is trained on. Since you have to provide good solutions for the training instances these can only be instances which are easy to solve by the MILP solvers. Your model should be evaluated on instances which are of larger dimension/harder than the training set.\n\nMinor comments:\n- The paragraph \"Metrics\" is not clear enough. What exactly do you mean hear by OBJ and what is BKS? Is BKS the best known solution after running the downstream solver? Why do you compare it to OBJ which I understand is the objective value of the solution predicted by the generative models?\n- The statement in abstract and conclusion that the primal gap is reduced by 41.34% is misleading. Gurobi is able to provide primal solutions nearly as good as your model in one hour.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)