  In the maximum coverage problem, we are given subsets $T_1, \ldots, T_m$ 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 $C(S) := \Big|\bigcup_{i \in S}
T_i\Big|$. It is a classic result that the greedy algorithm for this problem
achieves an optimal approximation ratio of $1-e^{-1}$.
  In this work we consider a generalization of this problem wherein an element
$a$ can contribute by an amount that depends on the number of times it is
covered. Given a concave, nondecreasing function $\varphi$, we define
$C^{\varphi}(S) := \sum_{a \in [n]}w_a\varphi(|S|_a)$, where $|S|_a = |\{i \in
S : a \in T_i\}|$. The standard maximum coverage problem corresponds to taking
$\varphi(j) = \min\{j,1\}$. For any such $\varphi$, we provide an efficient
algorithm that achieves an approximation ratio equal to the Poisson concavity
ratio of $\varphi$, defined by $\alpha_{\varphi} := \min_{x \in \mathbb{N}^*}
\frac{\mathbb{E}[\varphi(\text{Poi}(x))]}{\varphi(\mathbb{E}[\text{Poi}(x)])}$.
Complementing this approximation guarantee, we establish a matching NP-hardness
result when $\varphi$ grows in a sublinear way.
  As special cases, we improve the result of [Barman et al., IPCO, 2020] about
maximum multi-coverage, that was based on the unique games conjecture, and we
recover the result of [Dudycz et al., IJCAI, 2020] on multi-winner
approval-based voting for geometrically dominant rules. Our result goes beyond
these special cases and we illustrate it with applications to distributed
resource allocation problems, welfare maximization problems and approval-based
voting for general rules.
