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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 finda 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 anapproximation ratio of $(1-e^{-1})$ and there is a matching inapproximabilityresult. 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 wechange 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.  We study the maximum $\ell$-multi-coverage problem which naturallyinterpolates between these two extremes. In this problem, an element can becounted up to $\ell$ times but no more; hence, we consider maximizing thefunction $C^{(\ell)}(S) = \sum_{e \in [n]} \min\{\ell, |\{i \in S : e \inT_i\}| \}$, subject to the constraint $|S| \leq k$. Note that the case of $\ell= 1$ corresponds to the standard maximum coverage setting and $\ell = \infty$gives us a linear objective.  We develop an efficient approximation algorithm that achieves anapproximation ratio of $1 - \frac{\ell^{\ell}e^{-\ell}}{\ell!}$ for the$\ell$-multi-coverage problem. In particular, when $\ell = 2$, this factor is$1-2e^{-2} \approx 0.73$ and as $\ell$ grows the approximation ratio behaves as$1 - \frac{1}{\sqrt{2\pi \ell}}$. We also prove that this approximation ratiois tight, i.e., establish a matching hardness-of-approximation result, underthe Unique Games Conjecture.

Tight Approximation Bounds for Maximum Multi-Coverage