A Unified Approach to Approximating Partial Covering Problems

Abstract

An instance of the generalized partial cover problem consists of a ground set U and a family of subsets \({\mathcal{S}}\subseteq 2^{U}\) . Each element e∈U is associated with a profit p(e), whereas each subset \(S\in \mathcal{S}\) has a cost c(S). The objective is to find a minimum cost subcollection \(\mathcal{S}'\subseteq \mathcal{S}\) such that the combined profit of the elements covered by \(\mathcal{S}'\) is at least P, a specified profit bound. In the prize-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 π(e). The goal is to identify a subcollection \(\mathcal{S}'\subseteq \mathcal{S}\) that minimizes the cost of \(\mathcal{S}'\) 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.