[Paper Review] Symmetries in Linear and Integer Programs

Katrin Herr, Herr, K.|ArXiv.org|Aug 23, 2009

Optimization and Packing Problems8 references12 citations

TL;DR

This paper proposes a group-theoretic approach to exploit symmetries in linear and integer programs by reducing the problem dimension through symmetry-aware reformulation. It proves that an optimal solution always exists in the fixed point set of the symmetry group and introduces an algorithm to derive a lower-dimensional retract of the original problem, enabling efficient solution via dimensionality reduction based on orbit structure.

ABSTRACT

The notion of symmetry is defined in the context of Linear and Integer Programming. Symmetric linear and integer programs are studied from a group theoretical viewpoint. We show that for any linear program there exists an optimal solution in the fixed point set of its symmetry group. Using this result, we develop an algorithm that allows for reducing the dimension of any linear program having a non-trivial group of symmetries.

Motivation & Objective

• To formalize the role of symmetry in linear and integer programming using group theory.
• To address the computational challenge posed by symmetric integer programs, which hinder branch-and-bound efficiency due to redundant search paths.
• To develop a general, algebraic method for symmetry reduction that avoids reliance on branch-and-bound heuristics.
• To demonstrate that optimal solutions of symmetric linear programs lie within the fixed point set of their symmetry group.
• To provide a pre-processing algorithm that reduces the dimension of symmetric LPs by exploiting orbit structure.

Proposed method

• Define symmetries of a linear program as permutations of variables that preserve the constraint matrix and objective vector.
• Formalize the symmetry group as a subgroup of the symmetric group acting on the variable indices.
• Construct a retract of the original problem by projecting onto the fixed point space of the symmetry group using a matrix $ M_r $ derived from orbit representatives.
• Use the orbit-stabilizer theorem to determine the number of orbits, which determines the reduced problem dimension.
• Prove that the optimal solution of the original LP is recoverable from the solution of the lower-dimensional retract via the matrix $ M_r $.
• Apply the method iteratively if further symmetries are present in the reduced problem.

Experimental results

Research questions

• RQ1 Can symmetries in linear programs be systematically exploited to reduce problem dimension without altering the optimal solution?
• RQ2 Does every symmetric linear program admit an optimal solution that is invariant under its full symmetry group?
• RQ3 Can a general, algebraic method be developed to reduce symmetric LPs independent of specific optimization heuristics?
• RQ4 How does the number of orbits of the symmetry group relate to the dimensionality of the reduced problem?
• RQ5 To what extent can the symmetry reduction technique be applied iteratively to further simplify symmetric integer programs?

Key findings

• Every linear program with a non-trivial symmetry group admits an optimal solution that lies in the fixed point set of the group.
• The dimension of the reduced problem is determined solely by the number of orbits of the symmetry group, not by the group's internal structure.
• The optimal solution of the original LP can be reconstructed from the solution of the lower-dimensional retract using the matrix $ M_r $, which maps orbit representatives to original variables.
• The algorithm achieves dimensionality reduction by replacing the original problem with a symmetrically reduced version that preserves optimality.
• The method is general and pre-processing oriented, applicable to any LP with non-trivial symmetry, and does not depend on branch-and-bound heuristics.
• Even after reduction, the resulting problem may still exhibit symmetries, allowing for iterative application of the algorithm to achieve further simplification.