Integer Programming for Classifying Orthogonal Arrays
Abstract
Classifying orthogonal arrays is a well known important class of problems that asks for finding all nonisomorphic, nonnegative integer solutions to a class of systems of constraints. Solved instances are scarce. We develop two new methods based on finding all nonisomorphic solutions of two novel integer linear programming formulations for classifying all nonisomorphic OA(N,k,s,t) given a set of all nonisomorphic OA(N,k1,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 nonisomorphic solutions need to be enumerated by our methods. For each ILP, we determine the largest group of permutations that can be exploited with the branchandbound (B&B) with isomorphism pruning algorithm of Margot [Discrete Optim~4 (2007), 4062] without losing isomorphism classes of OA(N,k,2,t). Our contributions brought the classifications of all nonisomorphic 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.
 Publication:

arXiv eprints
 Pub Date:
 January 2015
 arXiv:
 arXiv:1501.02281
 Bibcode:
 2015arXiv150102281B
 Keywords:

 Mathematics  Combinatorics;
 05A15;
 62K05
 EPrint:
 21 pages