  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.
