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FMIP: Joint Continuous-Integer Flow For Mixed-Integer Linear Programming

Hongpei Li, Hui Yuan, Han Zhang, Jianghao Lin, Dongdong Ge, Mengdi Wang, Yinyu Ye

TL;DR

FMIP introduces the first generative framework that models the joint distribution of both integer and continuous variables for MILP, enabling holistic guidance during inference. By combining time-dependent joint flow with a problem-informed objective and constraint penalties, FMIP refines candidate solutions across multiple steps and improves solution quality. Empirical results across eight MILP benchmarks show a substantial reduction in primal gap (average of 41.34% relative) and demonstrated compatibility with diverse backbone graphs and downstream solvers. This approach offers a practical, solver-agnostic warm-start to accelerate real-world MILP 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.

FMIP: Joint Continuous-Integer Flow For Mixed-Integer Linear Programming

TL;DR

FMIP introduces the first generative framework that models the joint distribution of both integer and continuous variables for MILP, enabling holistic guidance during inference. By combining time-dependent joint flow with a problem-informed objective and constraint penalties, FMIP refines candidate solutions across multiple steps and improves solution quality. Empirical results across eight MILP benchmarks show a substantial reduction in primal gap (average of 41.34% relative) and demonstrated compatibility with diverse backbone graphs and downstream solvers. This approach offers a practical, solver-agnostic warm-start to accelerate real-world MILP 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.

Paper Structure

This paper contains 38 sections, 15 equations, 4 figures, 6 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Works
  3. Preliminaries
  4. MILP Definition and Graph Representation
  5. Flow Matching
  6. FMIP: Joint Continuous-Integer Flow for MILP
  7. Time-Dependent Graph Representation
  8. Joint Continuous-Integer Flow
  9. Joint Training Objective
  10. Sampling Process
  11. Holistic Guidance Mechanism
  12. Integration with Downstream Solvers
  13. Experiments
  14. Experiment Settings
  15. Datasets & Benchmarks
  16. ...and 23 more sections

Figures (4)

  • Figure 1: 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 2: The overall framework of our proposed FMIP.
  • 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.
  • Figure 4: Toy example of a MILP instance and its graph representation in Flow Matching.