Optimal shortening of covering arrays

A Covering Array (CA), denoted by CA(N ; t, k, v), is a matrix of size N ×k with entries from the set {0,1,2,...,v−1}, where in each submatrix of size N ×t appears each combination of symbols derived from vt, at least once. The Covering Arrays (CAs) are combinatorial structures that have applications in software testing. This paper defines the Problem of Optimal Shortening of Covering ARrays (OSCAR), gives its NP-Completeness proof and presents an exact and a greedy algorithms to solve it. The OSCAR problem is an optimization problem that for a given matrix M consists in finding a submatrix M′ that is close to be a CA. An algorithm that solves the OSCAR problem has application as an initialization function of a metaheuristic algorithm that constructs CAs. Our algorithms were tested on a benchmark formed by 20 instances of the OSCAR problem, derived from 12 CAs taken from the scientific literature. From the solutions of the 20 instances of the OSCAR problem, 12 were transformed into CAs through a previously reported metaheuristic algorithm for the construction of CAs.