Tight approximation bounds for maximum multi-coverage

Abstract

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 -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{ , |{i ∈ S : e ∈ T i }|}, 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 1 − e − ! for the -multi-coverage problem. In particular, when = 2, this factor is 1 − 2e −2 ≈ 0.73 and as grows the approximation ratio behaves as 1 − 1 √ 2π. We also prove that this approximation

Siting renewable power generation assets with combinatorial optimisation

“…From an algorithmic standpoint, the combinatorial optimisation problem studied in this paper is closely related to submodular maximisation [53] and coverage problems [40,19,38,7]. A number of negative results are known for these two classes of problems.…”

“…In other words, in the value oracle model, the problem is also hard to approximate. Inapproximability results are also known for several coverage problems, including maximum coverage [38] (which, given a collection of subsets of a ground set, consists in selecting k subsets so as to maximise the cardinality of their union), maximum -multi-coverage [7] (which can be viewed as a relaxed version of maximum coverage where an element can be counted up to times and is also a particular case of submodular maximisation subject to a cardinality constraint), set cover [40,16] and set multicover [19,57,58] (which, given a collection of subsets of a ground set, consist in selecting the smallest number of subsets so as to guarantee that each element of the ground set is covered at least once or multiple times, respectively). More precisely, unless P = NP, it is known that no polynomial-time algorithm with an approximation ratio better than 1 −1/e exists for the maximum coverage problem [25].…”

“…In addition, under the unique games conjecture [42], the best approximation ratio that any polynomial-time algorithm can achieve for maximum -multi-coverage is 1 − e − / ! [7]. Feige [25] also showed that set cover cannot be approximated efficiently below a threshold of (1 − o(1)) ln n (where n is the number of elements to be covered), unless NP has slightly superpolynomial-time algorithms.…”

“…Greedy algorithms also achieve logarithmic approximation ratios for both set cover [40,16] and set multicover [19,57]. Interestingly, for maximum -multi-coverage, the greedy algorithm does not have a tight approximation ratio [7]. Instead, the best algorithm combines a linear programming relaxation with the pipage rounding scheme [2,7].…”

“…Interestingly, for maximum -multi-coverage, the greedy algorithm does not have a tight approximation ratio [7]. Instead, the best algorithm combines a linear programming relaxation with the pipage rounding scheme [2,7]. Finally, it is worth mentioning that sophisticated heuristics lacking theoretical guarantees have also been used to tackle coverage problems, such as the greedy randomized adaptive search procedure (GRASP) of Resende [63].…”