Partial covering arrays: Algorithms and asymptotics

A covering arrayCA(N; t, k, v) is an N × k array with entries in {1,2,…,v}, for whichevery N × t subarray containseach t-tuple of {1,2,…,v}^tamong its rows. Covering arrays find application in interaction testing, including software and hardware testing, advanced materials development, and biological systems. A central question is to determine or bound CAN(t, k, v), the minimum number N of rows of a CA(N; t, k, v). The well known bound CAN(t, k, v) = O((t− 1)v^tlog k) is not too far from being asymptotically optimal. Sensible relaxations of the covering requirement arise when (1) the set {1,2,…,v}^tneed only be contained among the rows ofat least(1 − ϵ) (Formula present) of the N × t subarrays and (2) the rows ofevery N × t subarray need only contain a (large)subsetof {1,2,…,v}^t. In this paper, using probabilistic methods, significant improvements on the covering array upper bound are established for both relaxations, and for the conjunction of the two. In each case, a randomized algorithm constructs such arrays in expected polynomial time.