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

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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.

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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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Authors and Affiliations

  1. School of Computer Science and Technology, Shandong Jianzhu University, Jinan, 250101, People’s Republic of China

    Dongmei Zhang

  2. Beijing Institute for Scientific and Engineering Computing, Beijing University of Technology, Beijing, 100124, People’s Republic of China

    Dachuan Xu

  3. Shenzhen Institutes of Advanced Technology, Chinese Academy of Sciences, Shenzhen, 518055, People’s Republic of China

    Yishui Wang

  4. School of Computer Science and Technology, Shandong University, Jinan, 250101, People’s Republic of China

    Peng Zhang

  5. College of Applied Sciences, Beijing University of Technology, Beijing, 100124, People’s Republic of China

    Zhenning Zhang

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  1. Dongmei Zhang
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  4. Peng Zhang
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Correspondence to Dachuan Xu.

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This paper has been presented on the Global Optimization Conference (GOC-2017).

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Zhang, D., Xu, D., Wang, Y. et al. Local search approximation algorithms for the sum of squares facility location problems. J Glob Optim 74, 909–932 (2019). https://doi.org/10.1007/s10898-018-00733-2

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