Covering Array on the Cartesian Product of Hypergraphs

Abstract

Covering array (CA) on a hypergraph H is a combinatorial object used in interaction testing of a complex system modeled as H. Given a t-uniform hypergraph H and positive integer s, it is an array with a column for each vertex having entries from a finite set of cardinality s, such as \(\mathbb {Z}_s\), and the property that any set of t columns that correspond to vertices in a hyperedge covers all \(s^t\) ordered t-tuples from \(\mathbb {Z}_s^t\) at least once as a row. Minimizing the number of rows (size) of CA is important in industrial applications. Given a hypergraph H, a CA on H with the minimum size is called optimal. Determining the minimum size of CA on a hypergraph is NP-hard. We focus on constructions that make optimal covering arrays on large hypergraphs from smaller ones and discuss the construction method for optimal CA on the Cartesian product of a Cayley hypergraph with different families of hypergraphs. For a prime power \(q>2\), we present a polynomial-time approximation algorithm with approximation ratio \(\left( \Big \lceil \log _q\left( \frac{|V|}{3^{k-1}}\right) \Big \rceil \right) ^2\) for constructing covering array CA(n, H, q) on 3-uniform hypergraph \(H=(V,E)\) with \(k>1\) prime factors with respect to the Cartesian product.