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

Dongmei Zhang, Dachuan Xu (), Yishui Wang, Peng Zhang and Zhenning Zhang
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Dongmei Zhang: Shandong Jianzhu University
Dachuan Xu: Beijing University of Technology
Yishui Wang: Chinese Academy of Sciences
Peng Zhang: Shandong University
Zhenning Zhang: Beijing University of Technology

Journal of Global Optimization, 2019, vol. 74, issue 4, No 14, 909-932

Abstract: 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 $$ \mathcal {C} \subset \mathbb {R}^p$$ C ⊂ R p and a uniform center opening cost $$f>0$$ f > 0 . The goal is to open a finite center subset $$F \subset \mathbb {R}^p$$ F ⊂ 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 $$7.7721+ \epsilon $$ 7.7721 + ϵ and $$9+ \epsilon $$ 9 + ϵ for the continue and discrete SOS-FLP respectively.

Keywords: Approximation algorithm; K-means; Facility location; Local search (search for similar items in EconPapers)
Date: 2019
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Citations: View citations in EconPapers (1)

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DOI: 10.1007/s10898-018-00733-2

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