Partial Covering Arrays: Algorithms and Asymptotics - Sarkar, Kaushik; Colbourn, Charles J.; De Bonis, Annalisa; Vaccaro, Ugo (Theory of computing systems, 2018)
A covering array CA( N ; t , k , v ) is an N × k array with entries in {1,2,…, v }, for which every N × t subarray contains each t -tuple of {1,2,…, v } t among 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 t log k ) is not too far from being asymptotically optimal. Sensible relaxations of the covering requirement arise when (1) the set {1,2,…, v } t need only be contained among the rows of at least ( 1 − 𝜖 ) k t of the N × t subarrays and (2) the rows of every N × t subarray need only contain a (large) subset of {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.