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SymILO: A Symmetry-Aware Learning Framework for Integer Linear Optimization

Qian Chen, Tianjian Zhang, Linxin Yang, Qingyu Han, Akang Wang, Ruoyu Sun, Xiaodong Luo, Tsung-Hui Chang

TL;DR

SymILO introduces a symmetry-aware learning framework for ILPs by treating symmetry-induced permutation of optimal solutions as learnable operators and optimizing them jointly with a neural predictor via alternating minimization. The method reformulates the learning objective to minimize a symmetry-aware risk, $r_s$, and provides concrete optimization strategies for cyclic, dihedral, and symmetric groups, including a linear-assignment formulation for large symmetric groups. Empirical results on four ILP benchmarks with varying symmetries show that SymILO substantially improves downstream tasks (fix-and-optimize, local branching, node selection), achieving notable relative primal-gap reductions and significant performance gains over symmetry-agnostic baselines. The work demonstrates that explicitly leveraging intrinsic ILP symmetry can markedly enhance solution-prediction quality and downstream optimization efficiency, with practical implications for ML-assisted combinatorial optimization.

Abstract

Integer linear programs (ILPs) are commonly employed to model diverse practical problems such as scheduling and planning. Recently, machine learning techniques have been utilized to solve ILPs. A straightforward idea is to train a model via supervised learning, with an ILP as the input and an optimal solution as the label. An ILP is symmetric if its variables can be permuted without changing the problem structure, resulting in numerous equivalent and optimal solutions. Randomly selecting an optimal solution as the label can introduce variability in the training data, which may hinder the model from learning stable patterns. In this work, we incorporate the intrinsic symmetry of ILPs and propose a novel training framework called SymILO. Specifically, we modify the learning task by introducing solution permutation along with neural network weights as learnable parameters and then design an alternating algorithm to jointly optimize the loss function. We conduct extensive experiments on ILPs involving different symmetries and the computational results demonstrate that our symmetry-aware approach significantly outperforms three existing methods -- achieving $50.3\%$, $66.5\%$, and $45.4\%$ average improvements, respectively.

SymILO: A Symmetry-Aware Learning Framework for Integer Linear Optimization

TL;DR

SymILO introduces a symmetry-aware learning framework for ILPs by treating symmetry-induced permutation of optimal solutions as learnable operators and optimizing them jointly with a neural predictor via alternating minimization. The method reformulates the learning objective to minimize a symmetry-aware risk, , and provides concrete optimization strategies for cyclic, dihedral, and symmetric groups, including a linear-assignment formulation for large symmetric groups. Empirical results on four ILP benchmarks with varying symmetries show that SymILO substantially improves downstream tasks (fix-and-optimize, local branching, node selection), achieving notable relative primal-gap reductions and significant performance gains over symmetry-agnostic baselines. The work demonstrates that explicitly leveraging intrinsic ILP symmetry can markedly enhance solution-prediction quality and downstream optimization efficiency, with practical implications for ML-assisted combinatorial optimization.

Abstract

Integer linear programs (ILPs) are commonly employed to model diverse practical problems such as scheduling and planning. Recently, machine learning techniques have been utilized to solve ILPs. A straightforward idea is to train a model via supervised learning, with an ILP as the input and an optimal solution as the label. An ILP is symmetric if its variables can be permuted without changing the problem structure, resulting in numerous equivalent and optimal solutions. Randomly selecting an optimal solution as the label can introduce variability in the training data, which may hinder the model from learning stable patterns. In this work, we incorporate the intrinsic symmetry of ILPs and propose a novel training framework called SymILO. Specifically, we modify the learning task by introducing solution permutation along with neural network weights as learnable parameters and then design an alternating algorithm to jointly optimize the loss function. We conduct extensive experiments on ILPs involving different symmetries and the computational results demonstrate that our symmetry-aware approach significantly outperforms three existing methods -- achieving , , and average improvements, respectively.
Paper Structure (46 sections, 2 theorems, 25 equations, 8 figures, 6 tables, 1 algorithm)

This paper contains 46 sections, 2 theorems, 25 equations, 8 figures, 6 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related works
  3. Background and preliminaries
  4. ILPs
  5. Symmetry Group
  6. Symmetric group
  7. Cyclic and dihedral groups
  8. Classic supervised learning for solution prediction
  9. Methodology
  10. Reformulation of the learning task
  11. An alternating minimization algorithm
  12. Optimization over symmetry groups in (\ref{['eq:update_pi']})
  13. Cyclic and dihedral groups
  14. Symmetric group
  15. An overview of the SymILO framework
  16. ...and 31 more sections

Key Result

Proposition 4.1

Let $r^*$ and $r_s^*$ be the global minimal values of $\min_{\theta} r$ and $\min_{\theta,\pi} r_s$, respectively. Then, the following claims hold:

Figures (8)

  • Figure 1: Permutation examples with directed edges denoting mapping rules.
  • Figure 2: An overview of the SymILO framework.
  • Figure 3: The training and test risks v.s. the number of epochs on four benchmark problems.
  • Figure 4: Relative primal gaps at different times. Three downstream tasks, i.e., fix-and-optimize, local branching, and node selection, are evaluated with a time limit of 800 seconds. The results of the same downstream task use the same color. In addition, the relative primal gap of the Tuned CPLEX running on a single thread is also reported as the blue dashed line.
  • Figure 5: Equivalent solutions of Example \ref{['eg:bin_packing']}.
  • ...and 3 more figures

Theorems & Definitions (5)

  • Definition 3.1
  • Proposition 4.1
  • Proposition 4.2
  • Example B.0.1
  • Example B.0.2