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Local search approximation algorithms for the sum of squares facility location problems

  • Dongmei Zhang
  • Dachuan XuEmail author
  • Yishui Wang
  • Peng Zhang
  • Zhenning Zhang
Article
  • 56 Downloads

Abstract

In this paper, we study the sum of squares facility location problem (SOS-FLP) which is an important variant of k-means clustering. In the SOS-FLP, we are given a client set \( \mathcal {C} \subset \mathbb {R}^p\) and a uniform center opening cost \(f>0\). The goal is to open a finite center subset \(F \subset \mathbb {R}^p\) and to connect each client to the closest open center such that the total cost including center opening cost and the sum of squares of distances is minimized. The SOS-FLP is introduced firstly by Bandyapadhyay and Varadarajan (in: Proceedings of SoCG 2016, Article No. 14, pp 14:1–14:15, 2016) which present a PTAS for the fixed dimension case. Using local search and scaling techniques, we offer the first constant approximation algorithm for the SOS-FLP with general dimension. We further consider the discrete version of SOS-FLP, in which we are given a finite candidate center set with nonuniform opening cost comparing with the aforementioned (continue) SOS-FLP. By exploring the structures of local and optimal solutions, we claim that the approximation ratios are \(7.7721+ \epsilon \) and \(9+ \epsilon \) for the continue and discrete SOS-FLP respectively.

Keywords

Approximation algorithm K-means Facility location Local search 

Notes

Acknowledgements

We would like to thank the referee for the insightful and constructive comments. The research of the first author is supported by Doctoral Fund of Shandong Jianzhu University (No. XNBS1264) and Higher Educational Science and Technology Program of Shandong Province (No. J15LN22). The second author is supported by Natural Science Foundation of China (Nos. 11531014 and 11871081). The third author is supported by National Science Foundation of China (Nos. 61433012 and U1435215). The fourth author is supported by Natural Science Foundation of China (No. 61672323). The fifth author is supported by Beijing Excellent Talents Funding (No. 2014000020124G046).

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  • Dongmei Zhang
    • 1
  • Dachuan Xu
    • 2
    Email author
  • Yishui Wang
    • 3
  • Peng Zhang
    • 4
  • Zhenning Zhang
    • 5
  1. 1.School of Computer Science and TechnologyShandong Jianzhu UniversityJinanPeople’s Republic of China
  2. 2.Beijing Institute for Scientific and Engineering ComputingBeijing University of TechnologyBeijingPeople’s Republic of China
  3. 3.Shenzhen Institutes of Advanced TechnologyChinese Academy of SciencesShenzhenPeople’s Republic of China
  4. 4.School of Computer Science and TechnologyShandong UniversityJinanPeople’s Republic of China
  5. 5.College of Applied SciencesBeijing University of TechnologyBeijingPeople’s Republic of China

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