Submodular Clustering in Low Dimensions
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We study a clustering problem where the goal is to maximize the coverage of the input points by k chosen centers. Specifically, given a set of n points P ⊆ℝ^d, the goal is to pick k centers C ⊆ℝ^d that maximize the service ∑_p ∈ Pφ( 𝖽(p,C) ) to the points P, where 𝖽(p,C) is the distance of p to its nearest center in C, and φ is a non-increasing service function φ : ℝ^+ →ℝ^+. This includes problems of placing k base stations as to maximize the total bandwidth to the clients – indeed, the closer the client is to its nearest base station, the more data it can send/receive, and the target is to place k base stations so that the total bandwidth is maximized. We provide an n^ε^-O(d) time algorithm for this problem that achieves a (1-ε)-approximation. Notably, the runtime does not depend on the parameter k and it works for an arbitrary non-increasing service function φ : ℝ^+ →ℝ^+.
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