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.

中文翻译：

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凹面覆盖问题的严格近似保证

在最大覆盖问题中，我们给定了一个宇宙 $[n]$ 的子集 $T_1, \ldots, T_m$ 以及一个整数 $k$，目标是找到一个子集 $S \subseteq [m]$最大化 $C(S) 的大小 $k$ := \Big|\bigcup_{i \in S} T_i\Big|$。这个问题的贪心算法达到了$(1-e^{-1})$的最优逼近比，这是一个经典的结果。在这项工作中，我们考虑了这个问题的一般化，其中元素 $a$ 可以贡献的数量取决于它被覆盖的次数。给定一个凹的非递减函数 $\varphi$，我们定义 $C^{\varphi}(S) := \sum_{a \in [n]}w_a\varphi(|S|_a)$，其中 $|S |_a = |\{i \in S : a \in T_i\}|$。标准的最大覆盖问题对应于取 $\varphi(j) = \min\{j,1\}$。对于任何这样的 $\varphi$，我们提供了一种有效的算法，该算法可实现近似比等于 $\varphi$ 的泊松凹度比，定义为 $\alpha_{\varphi} := \min_{x \in \mathbb{N}^*} \frac{ \mathbb{E}[\varphi(\text{Poi}(x))]}{\varphi(\mathbb{E}[\text{Poi}(x)])}$。作为这种近似保证的补充，当 $\varphi$ 以次线性方式增长时，我们建立了一个匹配的 NP 硬度结果。作为特殊情况，我们改进了 [Barman et al., IPCO, 2020] 关于最大多重覆盖的结果，这是基于独特的博弈猜想，我们恢复了 [Dudycz et al., IJCAI, 2020] 的结果基于多赢家批准的几何优势规则投票。我们的结果超出了这些特殊情况，我们用分布式资源分配问题的应用来说明它，