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# Tight Approximation Guarantees for Concave Coverage Problems

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## Cite As

Siddharth Barman, Omar Fawzi, and Paul Fermé. Tight Approximation Guarantees for Concave Coverage Problems. In 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 187, pp. 9:1-9:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
https://doi.org/10.4230/LIPIcs.STACS.2021.9

## Abstract

In the 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 ⊆ [m] of size k that maximizes C(S) : = |⋃_{i ∈ S} T_i|. 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 φ, we define C^{φ}(S) := ∑_{a ∈ [n]}w_aφ(|S|_a), where |S|_a = |{i ∈ S : a ∈ T_i}|. The standard maximum coverage problem corresponds to taking φ(j) = min{j,1}. For any such φ, we provide an efficient algorithm that achieves an approximation ratio equal to the Poisson concavity ratio of φ, defined by α_{φ} : = min_{x ∈ ℕ^*} 𝔼[φ(Poi(x))] / φ(𝔼[Poi(x)]). Complementing this approximation guarantee, we establish a matching NP-hardness result when φ grows in a sublinear way. As special cases, we improve the result of [Siddharth Barman et al., 2020] about maximum multi-coverage, that was based on the unique games conjecture, and we recover the result of [Szymon Dudycz et al., 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.

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Packing and covering problems
• Theory of computation → Rounding techniques
• Theory of computation → Algorithmic game theory
##### Keywords
• Approximation Algorithms
• Coverage Problems
• Concave Function

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