The linear programming relaxation permutation symmetry group of an orthogonal array defining integer linear program

TL;DR

This paper characterizes the permutation symmetry groups of orthogonal array ILP formulations under LP relaxation, revealing new symmetries and providing explicit group isomorphisms for specific cases.

Contribution

It proves the isomorphism of the LP relaxation permutation symmetry group to specific groups in 2-level strength 1 and 2 cases, uncovering previously unknown symmetries.

Findings

LP relaxation symmetry group is isomorphic to S_2 wr S_k for strength 1

For strength 2, the group is isomorphic to S_2^k ⋊ S_{k+1} for k ≥ 4

New permutation symmetries are identified beyond the natural embedding

Abstract

There is always a natural embedding of $S_s\wr S_k$ into the linear programming (LP) relaxation permutation symmetry group of an orthogonal array integer linear programming (ILP) formulation with equality constraints. The point of this paper is to prove that in the $2$ level, strength $1$ case the LP relaxation permutation symmetry group of this formulation is isomorphic to $S_2\wr S_k$ for all $k$, and in the $2$ level, strength $2$ case it is isomorphic to $S_2^k\rtimes S_{k+1}$ for $k\geq 4$. The strength $2$ result reveals previously unknown permutation symmetries that can not be captured by the natural embedding of $S_2\wr S_k$. We also conjecture a complete characterization of the LP relaxation permutation symmetry group of the ILP formulation.