Barman, S and Fawzi, O and Ghoshal, S and Gürpınar, E (2022) Tight approximation bounds for maximum multi-coverage. In: Mathematical Programming, 192 (1-2). pp. 443-476.
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Abstract
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) : = | ∪ i∈STi|. 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∈ [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) = ∑ i∈S| Ti| , which can be easily maximized under a cardinality constraint. We study the maximum ℓ-multi-coverage problem which naturally interpolates between these two extremes. In this problem, an element can be counted up to ℓ times but no more; hence, we consider maximizing the function C(ℓ)(S) = ∑ e∈[n]min { ℓ, | { i∈ S: e∈ Ti} | } , subject to the constraint | S| ≤ k. Note that the case of ℓ= 1 corresponds to the standard maximum coverage setting and ℓ= ∞ gives us a linear objective. We develop an efficient approximation algorithm that achieves an approximation ratio of 1-ℓℓe-ℓℓ! for the ℓ-multi-coverage problem. In particular, when ℓ= 2 , this factor is 1 - 2 e- 2≈ 0.73 and as ℓ grows the approximation ratio behaves as 1-12πℓ. 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.
| Item Type: | Journal Article |
|---|---|
| Publication: | Mathematical Programming |
| Publisher: | Springer Science and Business Media Deutschland GmbH |
| Additional Information: | The copyright for this article belongs to the Authors. |
| Keywords: | Mathematical programming; Software engineering, Approximation bounds; Approximation ratios; Cardinality constraints; Combinatorial auction; Efficient approximation algorithms; Hardness of approximation; Linear objective functions; Unique games conjecture, Approximation algorithms |
| Department/Centre: | Division of Electrical Sciences > Computer Science & Automation |
| Date Deposited: | 15 Jun 2022 10:08 |
| Last Modified: | 15 Jun 2022 10:08 |
| URI: | https://eprints.iisc.ac.in/id/eprint/73786 |
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