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Learning Cut Generating Functions for Integer Programming

Hongyu Cheng, Amitabh Basu

TL;DR

This work tackles improving ILP solution efficiency by learning cut generating functions (CGFs) that generalize classical CG and GMI cuts within the branch-and-cut framework. It develops both one-dimensional and k-dimensional CGFs, provides rigorous sample complexity bounds via pseudo-dimension analysis, and extends to instance-dependent CGFs learned by neural networks mapping problem instances to CGF parameters. Theoretical contributions include explicit pseudo-dimension bounds and decomposition-based arguments that guarantee learnability for CGF classes, with practical demonstrations showing significant reductions in root-node tree size on Knapsack and Packing problems. The results highlight the potential of CGFs to outperform traditional cuts in practice, especially when tailored to specific instance distributions and even when using neural mappings to CGF parameters, thereby offering a scalable path to more effective data-driven ILP solvers.

Abstract

The branch-and-cut algorithm is the method of choice to solve large scale integer programming problems in practice. A key ingredient of branch-and-cut is the use of cutting planes which are derived constraints that reduce the search space for an optimal solution. Selecting effective cutting planes to produce small branch-and-cut trees is a critical challenge in the branch-and-cut algorithm. Recent advances have employed a data-driven approach to select optimal cutting planes from a parameterized family, aimed at reducing the branch-and-bound tree size (in expectation) for a given distribution of integer programming instances. We extend this idea to the selection of the best cut generating function (CGF), which is a tool in the integer programming literature for generating a wide variety of cutting planes that generalize the well-known Gomory Mixed-Integer (GMI) cutting planes. We provide rigorous sample complexity bounds for the selection of an effective CGF from certain parameterized families that provably performs well for any specified distribution on the problem instances. Our empirical results show that the selected CGF can outperform the GMI cuts for certain distributions. Additionally, we explore the sample complexity of using neural networks for instance-dependent CGF selection.

Learning Cut Generating Functions for Integer Programming

TL;DR

This work tackles improving ILP solution efficiency by learning cut generating functions (CGFs) that generalize classical CG and GMI cuts within the branch-and-cut framework. It develops both one-dimensional and k-dimensional CGFs, provides rigorous sample complexity bounds via pseudo-dimension analysis, and extends to instance-dependent CGFs learned by neural networks mapping problem instances to CGF parameters. Theoretical contributions include explicit pseudo-dimension bounds and decomposition-based arguments that guarantee learnability for CGF classes, with practical demonstrations showing significant reductions in root-node tree size on Knapsack and Packing problems. The results highlight the potential of CGFs to outperform traditional cuts in practice, especially when tailored to specific instance distributions and even when using neural mappings to CGF parameters, thereby offering a scalable path to more effective data-driven ILP solvers.

Abstract

The branch-and-cut algorithm is the method of choice to solve large scale integer programming problems in practice. A key ingredient of branch-and-cut is the use of cutting planes which are derived constraints that reduce the search space for an optimal solution. Selecting effective cutting planes to produce small branch-and-cut trees is a critical challenge in the branch-and-cut algorithm. Recent advances have employed a data-driven approach to select optimal cutting planes from a parameterized family, aimed at reducing the branch-and-bound tree size (in expectation) for a given distribution of integer programming instances. We extend this idea to the selection of the best cut generating function (CGF), which is a tool in the integer programming literature for generating a wide variety of cutting planes that generalize the well-known Gomory Mixed-Integer (GMI) cutting planes. We provide rigorous sample complexity bounds for the selection of an effective CGF from certain parameterized families that provably performs well for any specified distribution on the problem instances. Our empirical results show that the selected CGF can outperform the GMI cuts for certain distributions. Additionally, we explore the sample complexity of using neural networks for instance-dependent CGF selection.
Paper Structure (22 sections, 12 theorems, 50 equations, 3 figures, 1 table, 1 algorithm)

This paper contains 22 sections, 12 theorems, 50 equations, 3 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Formal setup of the problem
  3. Integer linear programming background
  4. Cut generation functions.
  5. Gomory fractional cut gomory1958outline:
  6. Gomory's mixed-integer cut gomory1960algorithm:
  7. Sample complexity of selecting cut generating functions
  8. One-dimensional cut generation functions
  9. The construction
  10. Pseudo-dimension bound
  11. $k$-dimensional cut generation functions
  12. The construction
  13. Computation and pseudo-dimension
  14. Learnability of instance-dependent cut generation functions
  15. Numerical experiments
  16. ...and 7 more sections

Key Result

Lemma 2.1

Let $h:\mathcal{I} \times \mathcal{P} \rightarrow \mathbb{Z}$, where $\mathcal{P} \subseteq \mathbb{R}^d$ for some natural number $d$. Suppose that for any fixed $I \in \mathcal{I}$, there exist at most $\Gamma$ rational functions, each given by the quotient of two polynomials of degree at most $a$

Figures (3)

  • Figure 1: Three cut generating functions on $[0,1)$, where $\pi_{f, s_1, s_2}^{p,q}$ is defined in \ref{['sec:LearnabilityOf1DimCGF']}.
  • Figure 2: Three examples of the one-dimensional cut generating functions $\pi_{f,s_1,s_2}^{p,q}$ on $[0,1)$.
  • Figure 3: Illustration of the proof of \ref{['thm:LearnabilityOfOptimalMu1Mu2']}.

Theorems & Definitions (20)

  • Lemma 2.1
  • Proposition 3.1
  • Theorem 3.2
  • Theorem 4.1
  • Proposition 4.2
  • Theorem 4.3
  • Theorem 5.1
  • Lemma A.1: Theorem 5.5 in matousek1999geometric, Lemma 17 in bartlett2019nearly, Lemma 3.3 in anthony1999neural, Proposition 2.4 in stanley2004introduction
  • proof : Proof of \ref{['lem:pseudo-dimension']}
  • Lemma A.2: Theorem 4.5 in balcan2022structural
  • ...and 10 more