Finding the dimension of a non-empty orthogonal array polytope

TL;DR

This paper uses representation theory to analyze the dimension of polytopes associated with orthogonal arrays, providing new restrictions and insights that challenge existing conjectures and extend understanding of their geometric properties.

Contribution

It introduces new theoretical restrictions on the possible dimensions of orthogonal array polytopes and examines the implications for related integer linear programs.

Findings

The polytope is full dimensional under certain conditions, but the conjecture may not always hold.

New restrictions on the feasible dimension values of the convex hull of feasible points.

Identifies symmetry-based restrictions in specific cases like n=2 and even s.

Abstract

By using representation theory, we reduce the size of the set of possible values for the dimension of the convex hull of all feasible points polytope of an orthogonal array (OA) defining integer linear program (ILP). Our results address the conjecture that if this polytope is non-empty, then it is full dimensional within the affine space where all the feasible points of the ILP's linear programming (LP) relaxation lie, raised by Appa et al., [On multi-index assignment polytopes, Linear Algebra and its Applications 416 (2-3) (2006), 224--241]. In particular, our theoretical results provide a sufficient condition for this polytope to be full dimensional within the LP relaxation affine space when it is non-empty. This sufficient condition implies all the known non-trivial values of the dimension of the $(k,s)$ assignment polytope. However, our results suggest that the conjecture mentioned above may not be true. More generally we provide previously unknown restrictions on the feasible values of the dimension of convex hull of all feasible points polytope of our OA defining ILP. We also determine all possible corresponding sets of equality constraints up to equivalence that can be implied by the integrality constraints of this ILP. Moreover, we find additional restrictions on the dimension of convex hull of feasible points and larger sets of corresponding equality constraints for the $n=2$ and even $s$ cases. These cases posses symmetries that do not necessarily exist in the $3\leq n$ or odd $s$ cases.