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Outer approximations of core points for integer programming

Naghmeh Shahverdi, Seyyedmahsa Banihashemi, David Bremner

TL;DR

The paper tackles the challenge of symmetry in integer linear programming by shifting from core-point enumeration to outer-approximation strategies that leverage orbit structure. It develops nonlinear, symmetry-driven constraints based on circulant and partial-circulant representations to bound core-point behavior, extending the approach from cyclic groups to direct-product and general permutation groups via subspace decompositions. Key contributions include new constraints for singular and non-singular circulant matrices, generalizations to partial circulants and direct products of cyclic groups, and algorithms that apply to diverse permutation-group structures. Computational experiments on synthetic symmetric ILPs show promising performance, suggesting these outer-approximation techniques can outperform traditional MILP solvers on problems with substantial cyclic symmetry and challenging feasibility structure.

Abstract

For several decades the dominant techniques for integer linear programming have been branching and cutting planes. Recently, several authors have developed core point methods for solving symmetric integer linear programs (ILPs). An integer point is called a core point if its orbit polytope is lattice-free. It has been shown that for symmetric ILPs, optimizing over the set of core points gives the same answer as considering the entire space. Existing core point techniques rely on the number of core points (or equivalence classes) being finite, which requires special symmetry groups. In this paper we develop some new methods for solving symmetric ILPs (based on outer approximations of core points) that do not depend on finiteness but are more efficient if the group has large disjoint cycles in its set of generators.

Outer approximations of core points for integer programming

TL;DR

The paper tackles the challenge of symmetry in integer linear programming by shifting from core-point enumeration to outer-approximation strategies that leverage orbit structure. It develops nonlinear, symmetry-driven constraints based on circulant and partial-circulant representations to bound core-point behavior, extending the approach from cyclic groups to direct-product and general permutation groups via subspace decompositions. Key contributions include new constraints for singular and non-singular circulant matrices, generalizations to partial circulants and direct products of cyclic groups, and algorithms that apply to diverse permutation-group structures. Computational experiments on synthetic symmetric ILPs show promising performance, suggesting these outer-approximation techniques can outperform traditional MILP solvers on problems with substantial cyclic symmetry and challenging feasibility structure.

Abstract

For several decades the dominant techniques for integer linear programming have been branching and cutting planes. Recently, several authors have developed core point methods for solving symmetric integer linear programs (ILPs). An integer point is called a core point if its orbit polytope is lattice-free. It has been shown that for symmetric ILPs, optimizing over the set of core points gives the same answer as considering the entire space. Existing core point techniques rely on the number of core points (or equivalence classes) being finite, which requires special symmetry groups. In this paper we develop some new methods for solving symmetric ILPs (based on outer approximations of core points) that do not depend on finiteness but are more efficient if the group has large disjoint cycles in its set of generators.

Paper Structure

This paper contains 15 sections, 10 theorems, 73 equations, 2 figures, 4 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Basic Definitions
  3. Circulant Matrices
  4. New Constraints for Singular Circulant Matrices
  5. New Constraints for Non-Singular Circulant Matrices
  6. Example: solving a symmetric ILP with Algorithm 1
  7. Partial-Circulant Matrices
  8. New Constraints for Direct Products of Cyclic Groups
  9. New Constraints For Permutation Groups
  10. Disjoint Cycles
  11. Product of disjoint cycles
  12. Combination of two of the above cases
  13. Non-disjoint cycles
  14. Computational Experiments
  15. Conclusions

Key Result

lemma 1

Let $c\in \mathbb{R}^{n}$ and suppose $\mathop{\mathrm{Cir}}\nolimits(c)$ is invertible. Then its inverse is $\mathop{\mathrm{Cir}}\nolimits( \widehat{T}(c))$, where $\widehat{T}(c)$ is defined as follows: Note that actually $\dfrac{1}{\langle c,\mathbf{1} \rangle} = \psi_{0}^{-1}.$

Figures (2)

  • Figure 1: Illustrating the proof of Lemma \ref{['n-layers']}.
  • Figure 2: Comparison of elapsed time for problem P2. Asterisks indicate timeout.

Theorems & Definitions (43)

  • definition 1
  • remark 1
  • definition 2: Core Points
  • definition 3
  • definition 4
  • definition 5: Equivalence relations
  • definition 6
  • definition 7
  • definition 8
  • definition 9
  • ...and 33 more