Mixed covering arrays on graphs

Abstract

Abstract:Covering arrays have applications in software, network and circuit testing. In this article, we consider a generalization of covering arrays that allows mixed alphabet sizes as well as a graph structure that specifies the pairwise interactions that need to be tested. Let k and n be positive integers, and let G be a graph with k vertices v 1 , v 2 , . . . , v k with respective vertex weights g 1 ≤ g 2 ≤ · · ·≤ g k . A mixed covering array on G, denoted by CA(n, G, k i=1 g i ), is an n × k array such that column i corresponds to v i , cells in column i are filled with elements from Z g i and every pair of columns i, j corresponding to an edge {v i , v j } in G has every possible pair from Z g i × Z g j appearing in some row. The number of rows in such array is called its size. Given a weighted graph G, a mixed covering array on G with minimum size is called optimal. In this article, we give upper and lower bounds on the size of mixed covering arrays on graphs based on graph homomorphisms. We provide constructions for covering arrays on graphs based on basic graph operations. In particular, we construct optimal mixed covering arrays on trees, cycles and bipartite graphs; the constructed optimal objects have the additional property of being nearly point balanced.

Variable strength covering arrays

“…Meagher and Stevens introduce and study these covering arrays on graphs in [66,68], and Meagher, Moura, and Zekaoui generalize to the mixed level case in [67]. We summarize some of their results here.…”

“…This bound is generalized to mixed-level covering arrays in [67]. There are strong connections between covering arrays on graphs, graph cores, and graph homomorphisms.…”

“…This motivates our presentation of qualitative independence hypergraphs in Section 3.2. This important family of hypergraphs, a generalization of qualitative independence graphs as given in [66,67,68], is connected directly to VCA via homomorphisms, and thus is of deep importance in the study of VCA. We establish hypergraph-theoretic results and properties of these hypergraphs, which provides insight into their structure.…”

“…Both Meagher et al [67] and Cheng [17] determine the covering array number in the case of G being a tree, cycle, or bipartite graph, showing that the lower bound as given in Equation 2.2 can be achieved.…”

“…In [67], for the mixed alphabet case, bounds for graphs of low chromatic number (χ(G) ∈ {2, 3, 4}) are given. Both Meagher et al [67] and Cheng [17] determine the covering array number in the case of G being a tree, cycle, or bipartite graph, showing that the lower bound as given in Equation 2.2 can be achieved.…”