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Learning to Remove Cuts in Integer Linear Programming

Pol Puigdemont, Stratis Skoulakis, Grigorios Chrysos, Volkan Cevher

TL;DR

It is demonstrated that in fundamental combinatorial optimization settings such cut removal policies can lead to significant improvements over both human-based and machine learning-guided cut addition policies even when implemented with simple models.

Abstract

Cutting plane methods are a fundamental approach for solving integer linear programs (ILPs). In each iteration of such methods, additional linear constraints (cuts) are introduced to the constraint set with the aim of excluding the previous fractional optimal solution while not affecting the optimal integer solution. In this work, we explore a novel approach within cutting plane methods: instead of only adding new cuts, we also consider the removal of previous cuts introduced at any of the preceding iterations of the method under a learnable parametric criteria. We demonstrate that in fundamental combinatorial optimization settings such cut removal policies can lead to significant improvements over both human-based and machine learning-guided cut addition policies even when implemented with simple models.

Learning to Remove Cuts in Integer Linear Programming

TL;DR

It is demonstrated that in fundamental combinatorial optimization settings such cut removal policies can lead to significant improvements over both human-based and machine learning-guided cut addition policies even when implemented with simple models.

Abstract

Cutting plane methods are a fundamental approach for solving integer linear programs (ILPs). In each iteration of such methods, additional linear constraints (cuts) are introduced to the constraint set with the aim of excluding the previous fractional optimal solution while not affecting the optimal integer solution. In this work, we explore a novel approach within cutting plane methods: instead of only adding new cuts, we also consider the removal of previous cuts introduced at any of the preceding iterations of the method under a learnable parametric criteria. We demonstrate that in fundamental combinatorial optimization settings such cut removal policies can lead to significant improvements over both human-based and machine learning-guided cut addition policies even when implemented with simple models.
Paper Structure (37 sections, 2 theorems, 22 equations, 8 figures, 4 tables, 2 algorithms)

This paper contains 37 sections, 2 theorems, 22 equations, 8 figures, 4 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Our Approach and Results
  3. Related Work
  4. Background and Preliminaries
  5. Evaluating Cut Selection Policies
  6. Cut Removal Policies
  7. Learning to Remove Cuts
  8. Training the Model
  9. Dataset Generation
  10. Experiments
  11. Evaluation Setup
  12. Datasets
  13. In Distribution Evaluation
  14. ILP Benchmarks
  15. Test Evaluation Metrics
  16. ...and 22 more sections

Key Result

Theorem 2.2

[gomory_cuts] Let an instance $(\mathcal{H},c)$ with a strictly fractional solution $x_{\mathrm{frac}}^\star$. Then there exists a set of hyperplanes $\mathcal{C}$ with $|\mathcal{C}| = |\mathcal{H}|$ such that any $(\alpha, \beta) \in \mathcal{C}$ is a cutting plane. $\mathcal{C}$ is also referred

Figures (8)

  • Figure 1: Visual representation of Algorithm \ref{['alg:CP2']}.
  • Figure 2: Mean IGC for the benchmark instances: We report the mean Integral Gap Closed (IGC) that measures quality of the solution with respect to the optimal integer solution (see Section \ref{['sec:background']} for details), a policy achieving larger IGC values with fewer iterations is better. The highlighted region represents the variance. Our cut removal algorithm outperforms or matches all cut addition methods in all benchmarks. For cut addition policies the $\textit{look-ahead}$ outperforms all of the others except in Packing where Neural Cut, this behavior matches the results showcased in the equivalent benchmark of paulus2022learning.
  • Figure 3: Mean IGC for out-of-distribution instances Our cut removal algorithm outperforms or matches all cut addition methods in all benchmarks. Our method shows a much stronger generalization ability onto the larger instances. The scaling improvement is especially accentuated in packing, bin packing and set cover.
  • Figure 4: Packing Cutpool Distributions
  • Figure 5: Bin Packing Cutpool Distributions
  • ...and 3 more figures

Theorems & Definitions (7)

  • Definition 2.1
  • Theorem 2.2
  • Remark 3.1
  • Corollary 3.2
  • proof
  • Remark 3.3
  • Remark 4.1