Folding Mixed-Integer Linear Programs and Reflection Symmetries

TL;DR

This paper extends the Dimension Reduction via Color Refinement (DRCR) method to handle reflection symmetries and applies it to mixed-integer linear programs, demonstrating significant computational benefits in solving large-scale problems.

Contribution

The authors extend DRCR to reflection symmetries and continuous columns, and incorporate affine totally unimodular decompositions for improved mixed-integer program reduction.

Findings

DRCR with reflection symmetries modestly reduces LP relaxation times.

DRCR significantly decreases solution times for MIP models in experiments.

Algorithms for symmetry detection are fast and scalable to large instances.

Abstract

For mixed-integer linear programming and linear programming it is well known that symmetries can have a negative impact on the performance of branch-and-bound and linear optimization algorithms. A common strategy to handle symmetries in linear programs is to reduce the dimension of the linear program by aggregating symmetric variables and solving a linear program of reduced dimension. In their work ``Dimension Reduction via Color Refinement'' (DRCR), Grohe, Kersting, Mladenov and Selman show that it is sufficient to run a fast color refinement algorithm to detect permutation symmetries and reduce the dimension of the linear program. We extend DRCR in two directions. First, we show that DRCR can be extended to reflection symmetries, which generalize permutation symmetries. Second, we show the folklore result that DRCR can be applied to the continuous columns of mixed-integer linear programs. In order to derive additional reductions on the integer variables we use affine totally unimodular decompositions to reformulate mixed-integer linear programs into mixed-integer linear programs with fewer integer variables. Computational experiments on MIPLIB 2017 collection set using SCIP 10 show that DRCR is an effective tool for handling symmetries. For the linear programming relaxations, DRCR with reflection symmetries yields a modest reduction in running time compared to the original DRCR procedure. For mixed-integer linear programming models, DRCR is very effective at reducing the solution time compared to the default configuration of SCIP. Moreover, the developed DRCR detection algorithms are fast and scale well to large problem instances.