Learning Cut Generating Functions for Integer Programming

TL;DR

This paper introduces a data-driven method for selecting cut generating functions in integer programming, providing theoretical guarantees and empirical evidence that it can outperform traditional GMI cuts for specific distributions.

Contribution

It extends the selection framework to cut generating functions, offers sample complexity bounds, and explores neural network-based selection methods.

Findings

Selected CGFs can outperform GMI cuts on certain distributions.

Theoretical sample complexity bounds are established for CGF selection.

Neural networks can be used for instance-dependent CGF selection.

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.