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

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⊂Rp 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⊂Rp 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.

[1]  Éva Tardos,et al.  Approximation algorithms for facility location problems (extended abstract) , 1997, STOC '97.

[2]  Samir Khuller,et al.  Greedy strikes back: improved facility location algorithms , 1998, SODA '98.

[3]  M MountDavid,et al.  A local search approximation algorithm for k-means clustering , 2004 .

[4]  Shi Li.  A 1.488 approximation algorithm for the uncapacitated facility location problem , 2013, Inf. Comput..

[5]  Cristina G. Fernandes,et al.  A systematic approach to bound factor-revealing LPs and its application to the metric and squared metric facility location problems , 2015, Math. Program..

[6]  Peter Nielsen,et al.  Instance Scale, Numerical Properties and Design of Metaheuristics: A Study for the Facility Location Problem , 2018, ArXiv.

[7]  David M. Mount,et al.  A local search approximation algorithm for k-means clustering , 2002, SCG '02.

[8]  Sudipto Guha,et al.  Improved combinatorial algorithms for the facility location and k-median problems , 1999, 40th Annual Symposium on Foundations of Computer Science (Cat. No.99CB37039).

[9]  Dongmei Zhang,et al.  A Local Search Approximation Algorithm for the k-means Problem with Penalties , 2017, COCOON.

[10]  Kamesh Munagala,et al.  Local Search Heuristics for k-Median and Facility Location Problems , 2004, SIAM J. Comput..

[11]  Meena Mahajan,et al.  The planar k-means problem is NP-hard , 2012, Theor. Comput. Sci..

[12]  Jirí Matousek,et al.  On Approximate Geometric k -Clustering , 2000, Discret. Comput. Geom..

[13]  David B. Shmoys,et al.  Approximation algorithms for facility location problems , 2000, APPROX.

[14]  Vijay V. Vazirani,et al.  Approximation algorithms for metric facility location and k-Median problems using the primal-dual schema and Lagrangian relaxation , 2001, JACM.

[15]  David P. Williamson,et al.  The Design of Approximation Algorithms , 2011 .

[16]  Dongmei Zhang,et al.  Local search approximation algorithms for the k-means problem with penalties , 2019, J. Comb. Optim..

[17]  Maxim Sviridenko,et al.  A Bi-Criteria Approximation Algorithm for k-Means , 2015, APPROX-RANDOM.