Tight Approximation Bounds for Maximum Multi-Coverage
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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]{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^ℓe^-ℓ/ℓ! for the ℓ-multi-coverage problem. In particular, when ℓ = 2, this factor is 1-2e^-2≈ 0.73 and as ℓ grows the approximation ratio behaves as 1 - 1/√(2πℓ). We also prove that this approximation ratio is tight, i.e., establish a matching hardness-of-approximation result, under the Unique Games Conjecture.
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