Graph-dependent Covering Arrays and LYM Inequalities

Abstract

The problems we study in this thesis are all related to covering arrays. Covering arrays are combinatorial designs, widely used as templates for efficient interaction-testing suites. They have connections to many areas including extremal set theory, design theory, and graph theory. We define and study several generalizations of covering arrays, and we develop a method which produces an infinite family of LYM inequalities for graph-intersecting collections. A common theme throughout is the dependence of these problems on graphs.

Our main contribution is an extremal method yielding LYM inequalities for $H$-intersecting collections, for every undirected graph $H$. Briefly, an $H$-intersecting collection is a collection of packings (or partitions) of an $n$-set in which the classes of every two distinct packings in the collection intersect according to the edges of $H$.

We define ``$F$-following" collections which, by definition, satisfy a LYM-like inequality that depends on the arcs of a ``follow" digraph $F$ and a permutation-counting technique. We fully characterize the correspondence between ``$F$-following" and ``$H$-intersecting" collections. This enables us to apply our inequalities to $H$-intersecting collections.

For each graph $H$, the corresponding inequality inherently bounds the maximum number of columns in a covering array with alphabet graph $H$. We use this feature to derive bounds for covering arrays with the alphabet graphs $S_3$ (the star on three vertices) and $\kvloop{3}$ ($K_3$ with loops). The latter improves a known bound for classical covering arrays of strength two.

We define covering arrays on column graphs and alphabet graphs which generalize covering arrays on graphs. The column graph encodes which pairs of columns must be $H$-intersecting, where $H$ is a given alphabet graph. Optimizing covering arrays on column graphs and alphabet graphs is equivalent to a graph-homomorphism problem to a suitable family of targets which generalize qualitative independence graphs. When $H$ is the two-vertex tournament, we give constructions and bounds for covering arrays on directed column graphs.

FOR arrays are the broadest generalization of covering arrays that we consider. We define FOR arrays to encompass testing applications where constraints must be considered, leading to forbidden, optional, and required interactions of any strength. We model these testing problems using a hypergraph. We investigate the existence of FOR arrays, the compatibility of their required interactions, critical systems, and binary relational systems that model the problem using homomorphisms.