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On the Connected Minimum Sum of Radii Problem

Authors Hyung-Chan An , Mong-Jen Kao

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LIPIcs.ISAAC.2024.7.pdf
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Subject Classification

ACM Subject Classification
  • Theory of computation → Facility location and clustering
Keywords
  • connected minimum sum of radii
  • minimum sum of radii
  • connected facility location
  • approximation algorithms
  • Steiner trees

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Abstract

In this paper, we consider the study for the connected minimum sum of radii problem. In this problem, we are given as input a metric defined on a set of facilities and clients, along with some cost parameters. The objective is to open a subset of facilities, assign every client to an open facilitiy, and connect open facilities using a Steiner tree so that the weighted (by cost parameters) sum of the maximum assignment distance of each facility and the Steiner tree cost is minimized. This problem introduces the min-sum radii objective, an objective function that is widely considered in the clustering literature, to the connected facility location problem, a well-studied network design/clustering problem. This problem is useful in communication network design on a shared medium, or energy optimization of mobile wireless chargers.
We present both a constant-factor approximation algorithm and hardness results for this problem. Our algorithm is based on rounding an LP relaxation that jointly models the min-sum of radii problem and the rooted Steiner tree problem. To round the solution we use a careful clustering procedure that guarantees that every open facility has a proxy client nearby. This allows a reinterpretation for part of the LP solution as a fractional rooted Steiner tree. Combined with a cost filtering technique, this yields a 5.542-approximation algorithm.

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Hyung-Chan An and Mong-Jen Kao. On the Connected Minimum Sum of Radii Problem. In 35th International Symposium on Algorithms and Computation (ISAAC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 322, pp. 7:1-7:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.ISAAC.2024.7

Author Details

Hyung-Chan An
  • Yonsei University, Seoul, Republic of Korea
Mong-Jen Kao
  • National Yang-Ming Chiao-Tung University, Hsinchu, Taiwan

Funding

This work was supported by Institute of Information & communications Technology Planning & Evaluation (IITP) grant funded by the Korea government(MSIT) (No. RS-2021-II212068, Artificial Intelligence Innovation Hub). This work was supported by the National Science and Technology Council (NSTC), Taiwan, grants 112-2628-E-A49-017-MY3, 113-2628-E-A49-017-MY3, and 113-2634-F-A49-001-MBK.

Acknowledgements

The authors made equal contributions. We thank the anonymous reviewers for their helpful comments.

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