In the classic 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 \(C(S) :=|\cup _{i \in S} T_i|\). 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. 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 \(|S| \le k\). 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 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 \(1-2e^{-2} \approx 0.73\) and as \(\ell \) grows the approximation ratio behaves as \(1 - \frac{1}{\sqrt{2\pi \ell }}\). We also prove that this approximation ratio is tight, i.e., establish a matching hardness-of-approximation result, under the Unique Games Conjecture. This problem is motivated by the question of finding a code that optimizes the list-decoding success probability for a given noisy channel. We show how the multi-coverage problem can be relevant in other contexts, such as combinatorial auctions.

在经典的最大覆盖问题中，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\(S \subseteq [m]\)的大小k最大化\(C(S) :=|\cup _{i \in S} T_i|\)。众所周知，这个问题的贪心算法达到了\(((1-e^{-1})\)的逼近比，并且存在匹配的不可逼近性结果。我们注意到，在最大覆盖问题中，如果一个元素\(e \in [n]\)被多个集合覆盖，它仍然只被计数一次。相比之下，如果我们改变问题并计算每个元素e覆盖多少次，我们就得到一个线性目标函数\(C^{(\infty )}(S) = \sum _{i \in S} |T_i|\)，它可以很容易地最大化在基数约束下。我们研究了最大\(\ell \) -multi-coverage 问题，它自然地在这两个极端之间进行了插值。在这个问题中，一个元素最多可以计数\(\ell \)次，但不能更多；因此，我们考虑最大化函数\(C^{(\ell )}(S) = \sum _{e \in [n]} \min \{\ell , |\{i \in S : e \in T_i\}| \}\)，受约束\(|S| \le k\)。请注意，\(\ell = 1\) 的情况对应于标准最大覆盖率设置和\(\ell = \infty \)给了我们一个线性目标。我们开发了一种有效的近似算法，该算法实现了\(\ell \)的近似比\(1 - \frac{\ell ^{\ell }e^{-\ell }}{\ell !} \) -多覆盖问题。特别是，当\(\ell = 2\) 时，这个因子是\(1-2e^{-2} \approx 0.73\)并且随着\(\ell \)增长，近似比率表现为\(1 - \压裂{1}{\sqrt{2\pi \ell }}\). 我们还证明了这个近似比是严格的，即在唯一游戏猜想下建立一个匹配的近似硬度结果。这个问题的动机是寻找一个代码，优化给定噪声信道的列表解码成功概率。我们展示了多覆盖问题如何与其他上下文相关，例如组合拍卖。