Tight Approximation Bounds for Maximum Multi-coverage

Abstract

In the classic maximum coverage problem, we are given subsets \(T_1, \dots , T_m\) of a universe [n] along with an integer k and the objective is to find a subset \(S \subseteq [m]\) of size k that maximizes \(C(S) := |\cup _{i \in S} T_i|\). It is well-known that the greedy algorithm for this problem achieves an approximation ratio of \((1-e^{-1})\) and there is a matching inapproximability result. We note that in the maximum coverage problem if an element \(e \in [n]\) is covered by several sets, it is still counted only once. By contrast, if we change the problem and count each element e as many times as it is covered, then we obtain a linear objective function, \(C^{(\infty )}(S) = \sum _{i \in S} |T_i|\), which can be easily maximized under a cardinality constraint.