Finding the dimension of a non-empty orthogonal array polytope

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 of an orthogonal array (OA) defining integer linear description (ILD). 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 ILD’s linear description (LD) relaxation lie, raised by Appa et al. (2006). In particular, our theoretical results provide a sufficient condition for this polytope to be full-dimensional within the LD 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 the convex hull of all feasible points of our OA defining ILD. We also determine all possible corresponding sets of equality constraints up to equivalence that can potentially be implied by the integrality constraints of this ILD. Moreover, we find additional restrictions on the dimension of the convex hull of all feasible points, and larger sets of corresponding equality constraints for the $n=2$ and even $s$ cases. Each of these cases possesses symmetries that do not necessarily exist in the $3\le n$ or odd $s$ cases. Finally, we discuss how to decrease the number of possible values for the dimension of the convex hull of all feasible points of an arbitrary ILD as well as generate sets of corresponding equality constraints with the zero right hand side. These are the only sets of zero right hand side equality constraints up to equivalence that can potentially be implied by the integrality constraints of the ILD.