Integer Programming for Classifying Orthogonal Arrays

Abstract

Classifying orthogonal arrays is a well known important class of problems that asks for finding all non-isomorphic, non-negative integer solutions to a class of systems of constraints. Solved instances are scarce. We develop two new methods based on finding all non-isomorphic solutions of two novel integer linear programming formulations for classifying all non-isomorphic OA(N,k,s,t) given a set of all non-isomorphic OA(N,k-1,s,t). We also establish the concept of orthogonal design equivalence of OA(N,k,2,t) to reduce the number of integer linear programs (ILPs) whose all non-isomorphic solutions need to be enumerated by our methods. For each ILP, we determine the largest group of permutations that can be exploited with the branch-and-bound (B&B) with isomorphism pruning algorithm of Margot [Discrete Optim~4 (2007), 40-62] without losing isomorphism classes of OA(N,k,2,t). Our contributions brought the classifications of all non-isomorphic OA(160,k,2,4) for k=9,10 and OA(176,k,2,4) for k=5,6,7,8,9,10 within computational reach. These are the smallest s=2, t=4 cases for which classification results are not available in the literature.