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

- 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).

## References

- 1.Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for \(k\)-median and facility location problems. SIAM J. Comput.
**33**, 544–562 (2004)MathSciNetCrossRefzbMATHGoogle Scholar - 2.Bandyapadhyay, S., Varadarajan, K.: On variants of \(k\)-means clustering. In: Proceedings of SoCG 2016, Article No. 14, pp. 14:1–14:15Google Scholar
- 3.Chalupa, D., Nielsen, P.: Instance scale, numerical properties and design of metaheuristics: a study for the facility location problem. ArXiv preprint arXiv:1801.03419 (2018)
- 4.Charikar, M., Guha, S.: Improved combinatorial algorithms for the facility location and \(k\)-median problems. In: Proceedings of FOCS, pp. 378–388 (1999)Google Scholar
- 5.Fernandes, C.G., Meira, L.A., Miyazawa, F.K., Pedrosa, L.L.: A systematic approach to bound factor-revealing LPs and its application to the metric and squared metric facility location problems. Math. Program.
**153**, 655–685 (2015)MathSciNetCrossRefzbMATHGoogle Scholar - 6.Guha, S., Khuller, S.: Greedy strikes back: improved facility location algorithms. J. Algorithms
**31**, 228–248 (1999)MathSciNetCrossRefzbMATHGoogle Scholar - 7.Jain, K., Vazirani, V.V.: Approximation algorithms for metric facility location and \(k\)-median problems using the primal–dual schema and Lagrangian relaxation. J. ACM
**48**, 274–296 (2001)MathSciNetCrossRefzbMATHGoogle Scholar - 8.Kanungoa, T., Mountb, D.M., Netanyahuc, N.S., Piatkoe, C.D., Silvermand, R., Wu, A.Y.: A local search approximation algorithm for \(k\)-means clustering. Comput. Geom. Theory Appl.
**2**, 89–112 (2004)MathSciNetCrossRefGoogle Scholar - 9.Li, S.: A \(1.488\) approximation algorithm for the uncapacitated facility location problem. Inf. Comput.
**222**, 45–58 (2013)MathSciNetCrossRefzbMATHGoogle Scholar - 10.Mahajan, M., Nimbhorkar, P., Varadarajan, K.: The planar \(k\)-means problem is NP-hard. Theor. Comput. Sci.
**442**, 13–21 (2012)MathSciNetCrossRefzbMATHGoogle Scholar - 11.Makarychev, K., Makarychev, Y., Sviridenko, M., Ward, J.: A bi-criteria approximation algorithm for \(k\)-means. In: Proceedings of APPROX/RONDOM’16, Article No. 14, pp. 14:1–14:20Google Scholar
- 12.Matoušek, J.: On approximate geometric \(k\)-clustering. Discrete Comput. Geom.
**24**, 61–84 (2000)MathSciNetCrossRefzbMATHGoogle Scholar - 13.Shmoys, D.B., Tardos,É., Aardal, K.: Approximation algorithms for facility location problems. In: Proceedings of STOC, pp. 265–274 (1997)Google Scholar
- 14.Ward, J.: Private Communication (2017)Google Scholar
- 15.Williamson, D.P., Shmoys, D.B.: The Design of Approximation Algorithms. Cambridge University Press, Cambridge (2011)CrossRefzbMATHGoogle Scholar
- 16.Zhang, D., Hao, C., Wu, C., Xu, D., Zhang, Z.: A local search approximation algorithm for the \(k\)-means problem with penalties. In: Proceedings of COCOON, pp. 568–574 (2017)Google Scholar
- 17.Zhang, P.: A new approximation algorithm for the \(k\)-facility location problem. Theor. Comput. Sci.
**384**, 126–135 (2007)MathSciNetCrossRefzbMATHGoogle Scholar