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Automatic Generation of Polynomial Symmetry Breaking Constraints

Madalina Erascu, Johannes Middeke

TL;DR

Symmetry in integer programming causes redundant search; the paper proposes a method to automatically generate non-linear symmetry-breaking constraints by selecting a base polynomial $h$ and a permutation group $G$ and creating inequalities $h(Px) - h(x) \le 0$. Applied to near half-capacity 0-1 bin packing, quadratic breakers outperform linear breakers and even built-in symmetry handling, especially at small sizes. This work is the first to automatically generate non-linear symmetry breakers and demonstrates a practical, scalable approach with potential applications beyond bin packing. The approach is lightweight to implement and integrates with standard solvers, offering a new avenue for enhancing solver performance on highly symmetric IPs.

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.

Automatic Generation of Polynomial Symmetry Breaking Constraints

TL;DR

Symmetry in integer programming causes redundant search; the paper proposes a method to automatically generate non-linear symmetry-breaking constraints by selecting a base polynomial and a permutation group and creating inequalities . Applied to near half-capacity 0-1 bin packing, quadratic breakers outperform linear breakers and even built-in symmetry handling, especially at small sizes. This work is the first to automatically generate non-linear symmetry breakers and demonstrates a practical, scalable approach with potential applications beyond bin packing. The approach is lightweight to implement and integrates with standard solvers, offering a new avenue for enhancing solver performance on highly symmetric IPs.

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.
Paper Structure (13 sections, 2 theorems, 15 equations, 6 figures, 3 tables)

This paper contains 13 sections, 2 theorems, 15 equations, 6 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Main Result
  3. Case Study
  4. Problem Description (Bin Packing Model)
  5. Benchmark Generation (Near Half-Capacity Families)
  6. Symmetry Group
  7. Symmetry Breakers Generation
  8. Experimental Results
  9. Experimental setup
  10. Symmetry breaker construction used in the experiments
  11. Interpretation of the results
  12. Conclusions and Future Work
  13. Detailed Per-Class Experimental Results

Key Result

theorem 1

Let $G$ be the symmetry group of eq:IP and let $R = \mathbb{R}[x]$ where $x = (x_1, \ldots, x_n)$. Then, for any $h(x) \in R$ and any $P \in G$, there is at least one optimal solution $y^* \in U^n$ which fulfils the inequality That means, $h(P x) - h(x) \leq 0$ can be added to the problem eq:IP as a symmetry breaker without losing the ability to find an optimal solution.

Figures (6)

  • Figure 1: The regions $2 y + x^2 - 2 x - y^2 \leq 0$ (left) and $y^3 - 3 y + 3 x - x^3 \leq 0$ (centre) and $y - x \leq 0$ (right) are shown in a darker shade. In each case, one optimal solution of the example problem is inside the region while the other one is not.
  • Figure 2: Combined results for all test cases (3, 5, 7, or 9 classes). The plot summarises the work units for each breaker shape and for each tested combinations of number of variables and permutations. The work units are scaled relative to the work units of the Baseline problem (i. e., the version without any breakers and with the same number of classes). The Baseline is shown as a solid purple line, and the relative timings for Gurobi's Default setting for each of the test cases are shown as dashed green lines.
  • Figure 3: Results for the three classes case only.
  • Figure 4: Results for the five classes case only.
  • Figure 5: Results for the seven classes case only.
  • ...and 1 more figures

Theorems & Definitions (10)

  • definition 1: Symmetry
  • definition 2: Symmetry Breaker
  • theorem 1
  • proof
  • Remark 2
  • Remark 3
  • definition 3: GroveBenson1985
  • theorem 4
  • proof : Proof of \ref{['thm:FundamentalRegion']}
  • Remark 5