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Constant-Factor Approximation Algorithms for the Parity-Constrained Facility Location Problem

Authors Kangsan Kim, Yongho Shin, Hyung-Chan An



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Author Details

Kangsan Kim
  • Devsisters Corp., Seoul, South Korea
Yongho Shin
  • Department of Computer Science, Yonsei University, Seoul, South Korea
Hyung-Chan An
  • Department of Computer Science, Yonsei University, Seoul, South Korea

Acknowledgements

We thank the anonymous reviewers of this paper for their helpful comments.

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Kangsan Kim, Yongho Shin, and Hyung-Chan An. Constant-Factor Approximation Algorithms for the Parity-Constrained Facility Location Problem. In 31st International Symposium on Algorithms and Computation (ISAAC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 181, pp. 21:1-21:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)
https://doi.org/10.4230/LIPIcs.ISAAC.2020.21

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 - odd, even, or 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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Approximation algorithms analysis
  • Theory of computation → Facility location and clustering
Keywords
  • Facility location problems
  • approximation algorithms
  • clustering problems
  • parity constraints

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