# Tight Approximation Bounds for Maximum Multi-coverage

Barman, S and Fawzi, O and Ghoshal, S and GÃ¼rpÄ±nar, E (2020) Tight Approximation Bounds for Maximum Multi-coverage. In: 21st International Conference on Integer Programming and Combinatorial Optimization, IPCO 2020, 8-10 Jun 2020, London, pp. 66-77.

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Official URL: https://doi.org/10.1007/978-3-030-45771-6_6

## Abstract

In the classic maximum coverage problem, we are given subsets Formula Presented of a universe n along with an integer k and the objective is to find a subset Formula Presented of size k that maximizes Formula Presented. It is well-known that the greedy algorithm for this problem achieves an approximation ratio of Formula Presented and there is a matching inapproximability result. We note that in the maximum coverage problem if an element Formula Presented 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, Formula Presented, which can be easily maximized under a cardinality constraint. We study the maximum Formula Presented-multi-coverage problem which naturally interpolates between these two extremes. In this problem, an element can be counted upÂ to Formula Presented times but no more; hence, we consider maximizing the function Formula Presented, subject to the constraint Formula Presented. Note that the case of Formula Presented corresponds to the standard maximum coverage setting and Formula Presented gives us a linear objective. We develop an efficient approximation algorithm that achieves an approximation ratio of Formula Presented for the Formula Presented-multi-coverage problem. In particular, when Formula Presented, this factor is Formula Presented and as Formula Presented grows the approximation ratio behaves as Formula Presented. 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. Â© 2020, Springer Nature Switzerland AG.

Item Type: Conference Paper Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) Lecture Notes in Computer Science Springer The copyright for this article belongs to Springer Combinatorial optimization; Integer programming, Approximation bounds; Approximation ratios; Cardinality constraints; Combinatorial auction; Efficient approximation algorithms; Hardness of approximation; Linear objective functions; Unique games conjecture, Approximation algorithms Division of Electrical Sciences > Computer Science & Automation 03 Nov 2021 10:14 03 Nov 2021 10:14 http://eprints.iisc.ac.in/id/eprint/65377