Tight Approximation Bounds for Maximum Multi-Coverage - Barman, Siddharth; Omar, Fawzi; Ghoshal, Suprovat; Gürpınar, Emirhan (arXiv.org, 2022)
In the classic maximum coverage problem, we are given subsets \(T_1, …, T_m\) 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, \(C^{(∞)}(S) = \sum_{i \in S} |T_i|\), which can be easily maximized under a cardinality constraint. We study the maximum \(\ell\)-multi-coverage problem which naturally interpolates between these two extremes. In this problem, an element can be counted up to \(\ell\) times but no more; hence, we consider maximizing the function \(C^{(\ell)}(S) = \sum_{e \in [n]} \min\{\ell, |\{i \in S : e \in T_i\}| \}\), subject to the constraint \(|S| ≤ k\). Note that the case of \(\ell = 1\) corresponds to the standard maximum coverage setting and \(\ell = ∞\) gives us a linear objective. We develop an efficient approximation algorithm that achieves an approximation ratio of \(1 - \frac{\ell^{\ell}e^{-\ell}}{\ell!}\) for the \(\ell\)-multi-coverage problem. In particular, when \(\ell = 2\), this factor is \(1-2e^{-2} ≈ 0.73\) and as \(\ell\) grows the approximation ratio behaves as \(1 - \frac{1}{\sqrt{2π \ell}}\). We also prove that this approximation ratio is tight, i.e., establish a matching hardness-of-approximation result, under the Unique Games Conjecture.