Abstract
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 each t-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.
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Research of KS and CJC was supported in part by the National Science Foundation under Grant No. 1421058.
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Sarkar, K., Colbourn, C.J., de Bonis, A., Vaccaro, U. (2016). Partial Covering Arrays: Algorithms and Asymptotics. In: Mäkinen, V., Puglisi, S., Salmela, L. (eds) Combinatorial Algorithms. IWOCA 2016. Lecture Notes in Computer Science(), vol 9843. Springer, Cham. https://doi.org/10.1007/978-3-319-44543-4_34
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