# Upper bounds on the sizes of variable strength covering arrays using the Lov\'{a}sz local lemma

## TL;DR

This paper derives upper bounds on the size of variable strength covering arrays using the Lovász local lemma, extending probabilistic methods to more general hypergraph-based coverage requirements in software testing.

## Contribution

It introduces a probabilistic approach to bound the size of variable strength covering arrays based on hypergraph parameters, generalizing previous results for standard covering arrays.

## Key findings

- Upper bounds depend on hypergraph class and properties.
- Comparison with greedy algorithms shows varying effectiveness.
- Certain hypergraph structures are more suitable for Lovász local lemma techniques.

## Abstract

Covering arrays are generalizations of orthogonal arrays that have been widely studied and are used in software testing. The probabilistic method has been employed to derive upper bounds on the sizes of minimum covering arrays and give asymptotic upper bounds that are logarithmic on the number of columns of the array. This corresponds to test suites with a desired level of coverage of the parameter space where we guarantee the number of test cases is logarithmic on the number of parameters of the system. In this paper, we study variable strength covering arrays, a generalization of covering arrays that uses a hypergraph to specify the sets of columns where coverage is required; (standard) covering arrays is the special case where coverage is required for all sets of columns of a fixed size $t$, its strength. We use the probabilistic method to obtain upper bounds on the number of rows of a variable strength covering array, given in terms of parameters of the hypergraph. We then compare this upper bound with another one given by a density-based greedy algorithm on different types of hypergraph such as $t$-designs, cyclic consecutive hypergraphs, planar triangulation hypergraphs, and a more specific hypergraph given by a clique of higher strength on top of a "base strength". The conclusions are dependent on the class of hypergraph, and we discuss specific characteristics of the hypergraphs which are more amenable to using different versions of the Lov\'{a}sz local lemma.