Mixed Covering Arrays on 3-Uniform Hypergraphs

Abstract

Covering arrays are combinatorial objects that have been successfully applied in the design of test suites for testing systems such as software, circuits and networks, where failures can be caused by the interaction between their parameters. In this paper, we perform a new generalization of covering arrays called covering arrays on 3-uniform hypergraphs. Let $n, k$ be positive integers with $k\geq 3$. Three vectors $x\in \mathbb Z_{g_1}^n$, $y\in \mathbb Z_{g_2}^n$, $z\in \mathbb Z_{g_3}^n$ are {\it 3-qualitatively independent} if for any triplet $(a, b, c) \in \mathbb Z_{g_1}\,\times\, \mathbb Z_{g_2}\,\times\,\mathbb Z_{g_3}$, there exists an index $ j\in \lbrace 1, 2,...,n \rbrace $ such that $( x(j), y(j), z(j)) = (a, b, c)$. Let $H$ be a 3-uniform hypergraph with $k$ vertices $v_1,v_2,\ldots,v_k$ with respective vertex weights $g_1,g_2,\ldots,g_k$. A mixed covering array on $H$, denoted by $3-CA(n,H, \prod_{i=1}^{k}g_{i})$, is a $k\times n$ array such that row $i$ corresponds to vertex $v_i$, entries in row $i$ are from $Z_{g_i}$; and if $\{v_x,v_y,v_z\}$ is a hyperedge in $H$, then the rows $x,y,z$ are 3-qualitatively independent. The parameter $n$ is called the size of the array. Given a weighted 3-uniform hypergraph $H$, a mixed covering array on $H$ with minimum size is called optimal. We outline necessary background in the theory of hypergraphs that is relevant to the study of covering arrays on hypergraphs. In this article, we introduce five basic hypergraph operations to construct optimal mixed covering arrays on hypergraphs. Using these operations, we provide constructions for optimal mixed covering arrays on $\alpha$-acyclic 3-uniform hypergraphs, conformal 3-uniform hypertrees having a binary tree as host tree, and on some specific 3-uniform cycle hypergraphs.