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 $𝒮\subseteq 2^U$ . Each element e∈U is associated with a profit p(e), whereas each subset $S\in 𝒮$ has a cost c(S). The objective is to find a minimum cost subcollection $𝒮\prime \subseteq 𝒮$ such that the combined profit of the elements covered by $𝒮\prime$ 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 $𝒮\prime \subseteq 𝒮$ that minimizes the cost of $𝒮\prime$ 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.