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Constant-Factor Approximation Algorithms for Parity-Constrained Facility Location and k-Center

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Abstract

Facility location is a prominent optimization problem that has inspired a large quantity of both theoretical and practical studies in combinatorial optimization. Although the problem has been investigated under various settings reflecting typical structures within the optimization problems of practical interest, little is known on how the problem behaves in conjunction with parity constraints. This shortfall of understanding was rather discouraging when we consider the central role of parity in the field of combinatorics. In this paper, we present the first constant-factor approximation algorithm for the facility location problem with parity constraints. We are given as the input a metric on a set of facilities and clients, the opening cost of each facility, and the parity requirement–\(\textsf{odd}\), \(\textsf{even}\), or \(\textsf{unconstrained}\)–of every facility in this problem. The objective is to open a subset of facilities and assign every client to an open facility so as to minimize the sum of the total opening costs and the assignment distances, but subject to the condition that the number of clients assigned to each open facility must have the same parity as its requirement. Although the unconstrained facility location problem as a relaxation for this parity-constrained generalization has unbounded gap, we demonstrate that it yields a structured solution whose parity violation can be corrected at small cost. This correction is prescribed by a T-join on an auxiliary graph constructed by the algorithm. This auxiliary graph does not satisfy the triangle inequality, but we show that a carefully chosen set of shortcutting operations leads to a cheap and sparse T-join. Finally, we bound the correction cost by exhibiting a combinatorial multi-step construction of an upper bound. We also consider the parity-constrained k-center problem, the bottleneck optimization variant of parity-constrained facility location. We present the first constant-factor approximation algorithm also for this problem.

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Notes

  1. In Sect. 4.5, we show that a more careful analysis yields the approximation ratio of 5.813.

  2. Consider an instance with two pairs of an even-constrained facility and a client, where the distance within each pair is zero and one across. Both opening costs are zeroes.

  3. We obtain a 6-approximation algorithm that runs in \(O(|V|^2\log |V|)\) time; whilst this approximation ratio is obtained under the standard setting where there is no distinction between facilities and clients, we can easily extend the analysis to obtain an O(1)-approximation algorithm even with the distinction.

  4. Assignment costs are sometimes defined only between facilities and clients. In this “bipartite” case, the domain of c will be defined as \(F\times D\) instead. These two definitions, however, are equivalent, since we can deduce inter-facility (and inter-client) distances by computing the metric closure of the given “bipartite” assignment cost.

  5. We remark that a more careful analysis yields a tighter bound of \(2 c(\sigma _\mathcal {I}) + 2 c(\sigma _\mathcal {O}) + f(S_\mathcal {O}) \), although we present the current proof in favor of simplicity.

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Acknowledgements

We thank the anonymous referees of this paper and its conference version for their helpful comments.

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  1. Devsisters Corp., Seoul, South Korea

    Kangsan Kim

  2. Department of Computer Science, Yonsei University, Seoul, South Korea

    Yongho Shin & Hyung-Chan An

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This paper extends its preliminary version [31] that was presented at ISAAC 2020. Sections 5 and 4.5 were newly added in particular.

Part of this research was conducted while K. Kim was at Yonsei University. Part of this research was conducted while H.-C. An was visiting Cornell University. This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. NRF-2019R1C1C1008934). This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. NRF-2016R1C1B1012910). This research was supported by the Yonsei University Research Fund of 2018-22-0093. This work was partly supported by Institute of Information & communications Technology Planning & Evaluation (IITP) grant funded by the Korea government (MSIT) (No.2021-0-02068, Artificial Intelligence Innovation Hub). This research was supported by the Yonsei Signature Research Cluster Program of 2022 (2022-22-0002).

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Kim, K., Shin, Y. & An, HC. Constant-Factor Approximation Algorithms for Parity-Constrained Facility Location and k-Center. Algorithmica 85, 1883–1911 (2023). https://doi.org/10.1007/s00453-022-01060-5

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