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Submodular Clustering in Low Dimensions

Authors Arturs Backurs, Sariel Har-Peled

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Subject Classification

ACM Subject Classification
  • Theory of computation → Facility location and clustering
  • Theory of computation → Computational geometry
Keywords
  • clustering
  • covering
  • PTAS

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Abstract

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∈P}φ(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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Arturs Backurs and Sariel Har-Peled. Submodular Clustering in Low Dimensions. In 17th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 162, pp. 8:1-8:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020) https://doi.org/10.4230/LIPIcs.SWAT.2020.8

Author Details

Arturs Backurs
  • Toyota Technological Institute at Chicago, Il, USA
Sariel Har-Peled
  • University of Illinois at Urbana-Champaign, Il, USA

Funding

  • Har-Peled, Sariel: Work on this paper was partially supported by a NSF AF awards CCF-1907400 and CCF-1421231.

References

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