Partial Covering Arrays: Algorithms and Asymptotics - Sarkar, Kaushik; Colbourn, Charles J.; de Bonis, Annalisa; Vaccaro, Ugo (Combinatorial Algorithms, 2016)
A covering array $$\mathsf {CA}(N;t,k,v)$$ is an $$N\times k$$ array with entries in $$\{1, 2, \ldots , v\}$$ , for which every $$N\times t$$ subarray contains eacht-tuple of $$\{1, 2, \ldots , v\}^t$$ 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 $$\mathsf {CAN}(t,k,v)$$ , the minimum number N of rows of a $$\mathsf {CA}(N;t,k,v)$$ . The well known bound $$\mathsf {CAN}(t,k,v)=O((t-1)v^t\log k)$$ is not too far from being asymptotically optimal. Sensible relaxations of the covering requirement arise when (1) the set $$\{1, 2, \ldots , v\}^t$$ need only be contained among the rows of at least $$(1-\epsilon )\left( {\begin{array}{c}k\\ t\end{array}}\right) $$ of the $$N\times t$$ subarrays and (2) the rows of every $$N\times t$$ subarray need only contain a (large) subset of $$\{1, 2, \ldots , v\}^t$$ . 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.