Covering Array on the Cartesian Product of Hypergraphs

• Published in: Graphs and combinatorics Vol. 40; no. 4; p. 87
• Main Authors: Akhtar, Yasmeen, Maity, Soumen
• Format: Journal Article
• Language: English
• Published: Tokyo Springer Japan 01.08.2024; Springer Nature B.V
• Subjects: Apexes; Approximation; Arrays; Cartesian coordinates; Combinatorial analysis; Combinatorics; Complex systems; Engineering Design; Graph theory; Graphs; Industrial applications; Mathematics; Mathematics and Statistics; Original Paper; Polynomials; Covering array; Approximation algorithm; Cayley hypergraph; Cartesian product
• ISSN: 0911-0119; 1435-5914
• DOI: 10.1007/s00373-024-02813-5

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 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 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 ⌈ log q | V | 3 k - 1 ⌉ 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.