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Communication Dans Un Congrès Année : 2020

Tight Approximation Bounds for Maximum Multi-Coverage

Résumé

In the classic maximum coverage problem, we are given subsets T1,,Tm 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):=|iSTi|. It is well-known that the greedy algorithm for this problem achieves an approximation ratio of (1e1) and there is a matching inapproximability result. We note that in the maximum coverage problem if an element e[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()(S)=iS|Ti|, which can be easily maximized under a cardinality constraint. We study the maximum -multi-coverage problem which naturally interpolates between these two extremes. In this problem, an element can be counted up to times but no more; hence, we consider maximizing the function C()(S)=e[n]min{,|{iS:eTi}|}, subject to the constraint |S|k. Note that the case of =1 corresponds to the standard maximum coverage setting and = gives us a linear objective. We develop an efficient approximation algorithm that achieves an approximation ratio of 1e! for the -multi-coverage problem. In particular, when =2, this factor is 12e20.73 and as grows the approximation ratio behaves as 112π. We also prove that this approximation ratio is tight, i.e., establish a matching hardness-of-approximation result, under the Unique Games Conjecture.

Dates et versions

hal-03130103 , version 1 (03-02-2021)

Identifiants

Citer

Siddharth Barman, Omar Fawzi, Emirhan Gürpınar, Suprovat Ghoshal. Tight Approximation Bounds for Maximum Multi-Coverage. IPCO 2020 - 21st International Conference on Integer Programming and Combinatorial Optimization, Jun 2020, London, United Kingdom. pp.66-77, ⟨10.1007/978-3-030-45771-6_6⟩. ⟨hal-03130103⟩
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