Tight approximation bounds for maximum multi-coverage

• Published in: Mathematical programming Vol. 192; no. 1-2; pp. 443 - 476
• Main Authors: Barman, Siddharth, Fawzi, Omar, Ghoshal, Suprovat, Gürpınar, Emirhan
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.03.2022; Springer; Springer Nature B.V
• Subjects: Algorithms; Approximation; Calculus of Variations and Optimal Control; Optimization; Combinatorial analysis; Combinatorics; Decoding; Full Length Paper; Greedy algorithms; Matching; Mathematical analysis; Mathematical and Computational Physics; Mathematical Methods in Physics; Mathematics; Mathematics and Statistics; Mathematics of Computing; Numerical Analysis; Ratios; Theoretical; Randomized rounding; Maximum coverage; 68W25; Approximation algorithm; Convex order
• ISSN: 0025-5610; 1436-4646
• DOI: 10.1007/s10107-021-01677-4

In the classic 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 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 | T i | , 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 ∈ T i } | } , 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 - ℓ ℓ e - ℓ ℓ ! 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 - 1 2 π ℓ . 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.