A Unified Approach to Approximating Partial Covering Problems

An instance of the generalized partial cover problem consists of a ground set U and a family of subsets S⊆2U. Each element e ∈ U is associated with a profit p(e), whereas each subset S∈S has a cost c(S). The objective is to find a minimum cost subcollection S'⊆S such that the combined profit of the elements covered by S' is at least p, a specified profit bound. In the Drize-collecting version of this problem, there is no strict requirement to cover any element; however, if the subsets we pick leave an element e ∈ U uncovered, we incur a penalty of n(e). The goal is to identify a subcollection S'⊆S that minimizes the cost of S' plus the penalties of uncovered elements. Although problem-specific connections between the partial cover and the prize-collecting variants of a given covering problem have been explored and exploited, a more general connection remained open. The main contribution of this paper is to establish a formal relationship between these two variants. As a result, we present a unified framework for approximating problems that can be formulated or interpreted as special cases of generalized partial cover. We demonstrate the applicability of our method on a diverse collection of covering problems, for some of which we obtain the first non-trivial approximability results.