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Recent Developments in Approximation Algorithms for Facility Location and Clustering Problems

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Combinatorial Optimization and Graph Algorithms

Abstract

We survey approximation algorithms for facility location and clustering problems, focusing on the recent developments. In particular, we review two algorithmic methodologies that have successfully lead to the current best approximation guarantees known: local search and linear programming based methods.

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Notes

  1. 1.

    This factor was improved to \((5+\varepsilon )\) in the journal version [31] of the same paper.

  2. 2.

    This choice of constant must be large enough to offset the choice of 9 in the lemma’s statement.

  3. 3.

    We have that the opening cost of \(\bar{S}\) is greater than a constant fraction of the opening cost of S; moreover, this inequality has a small slack that is proportional to the per facility connection cost of \(\bar{S}\).

  4. 4.

    This, of course, is true only for those clients that are near i, the center of the ball.

  5. 5.

    The comparison has a slight error (see Note 4), which is charged against the slack described in Note 3. This balancing, along with the choice of constant in Note 2, determines the overall approximation ratio of 9.

  6. 6.

    We remark that the conference version of this paper claimed a guarantee of \(2.611+\varepsilon \) where the additional improvement came from an improved Lagrangian multiplier preserving algorithm for facility location. The up-to-date arXiv version of the same paper [11] however noted that this part of their improvement unfortunately contained an error, changing the final approximation guarantee to \(2.675+\varepsilon \).

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Acknowledgements

We thank the anonymous reviewer of this article for the helpful comments. This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIP) (No. 2016R1C1B1012910), a Yonsei University New Faculty Seed Grant, and ERC Starting Grant 335288-OptApprox.

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Authors and Affiliations

  1. Department of Computer Science, Yonsei University, Seoul, 03722, Korea

    Hyung-Chan An

  2. School of Computer and Communication Sciences, École Polytechnique Fédérale de Lausanne, 1015, Lausanne, Switzerland

    Ola Svensson

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Correspondence to Hyung-Chan An .

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  1. National Institute of Informatics , Tokyo, Japan

    Takuro Fukunaga

  2. National Institute of Informatics , Tokyo, Japan

    Ken-ichi Kawarabayashi

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An, HC., Svensson, O. (2017). Recent Developments in Approximation Algorithms for Facility Location and Clustering Problems. In: Fukunaga, T., Kawarabayashi, Ki. (eds) Combinatorial Optimization and Graph Algorithms. Springer, Singapore. https://doi.org/10.1007/978-981-10-6147-9_1

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  • DOI: https://doi.org/10.1007/978-981-10-6147-9_1

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