Automatic Generation of Polynomial Symmetry Breaking Constraints

TL;DR

This paper introduces an algebraic method for automatically generating polynomial symmetry breaking constraints to reduce redundant search in integer programming, demonstrated on bin packing problems.

Contribution

It presents a novel algebraic approach to generate polynomial symmetry breakers based on a base polynomial and permutation group, adaptable to various problems.

Findings

Simple symmetry breakers significantly reduce solving time.

Combining few variables and permutations yields the best performance.

Method is easily implementable in symbolic computation software.

Abstract

Symmetry in integer programming causes redundant search and is often handled with symmetry breaking constraints that remove as many equivalent solutions as possible. We propose an algebraic method which allows to generate a random family of polynomial inequalities which can be used as symmetry breakers. The method requires as input an arbitrary base polynomial and a group of permutations which is specific to the integer program. The computations can be easily carried out in any major symbolic computation software. In order to test our approach, we describe a case study on near half-capacity 0-1 bin packing instances which exhibit substantial symmetries. We statically generate random quadratic breakers and add them to a baseline integer programming problem which we then solve with Gurobi. It turns out that simple symmetry breakers, especially combining few variables and permutations, most consistently reduce work time.