Partial Covering Arrays: Algorithms and Asymptotics

• Published in: Theory of computing systems Vol. 62; no. 6; pp. 1470 - 1489
• Main Authors: Sarkar, Kaushik, Colbourn, Charles J., De Bonis, Annalisa, Vaccaro, Ugo
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.08.2018; Springer Nature B.V
• Subjects: Algorithms; Arrays; Biological materials; Computer Science; Probabilistic methods; Set theory; Special Issue on Combinatorial Algorithms; Theory of Computation; Upper bounds; Combinatorial design; Covering arrays; Partial covering arrays; Software interaction testing; Orthogonal arrays
• ISSN: 1432-4350; 1433-0490
• DOI: 10.1007/s00224-017-9782-9

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.