Journal of Combinatorial Optimization

, Volume 35, Issue 4, pp 1168–1184 | Cite as

A local search approximation algorithm for a squared metric k-facility location problem

  • Dongmei Zhang
  • Dachuan Xu
  • Yishui Wang
  • Peng Zhang
  • Zhenning Zhang
Article
  • 34 Downloads

Abstract

In this paper, we introduce a squared metric k-facility location problem (SM-k-FLP) which is a generalization of the squared metric facility location problem and k-facility location problem (k-FLP). In the SM-k-FLP, we are given a client set \(\mathcal {C}\) and a facility set \(\mathcal {F} \) from a metric space, a facility opening cost \(f_i \ge 0\) for each \( i \in \mathcal {F}\), and an integer k. The goal is to open a facility subset \(F \subseteq \mathcal {F}\) with \( |F| \le k\) and to connect each client to the nearest open facility such that the total cost (including facility opening cost and the sum of squares of distances) is minimized. Using local search and scaling techniques, we offer a constant approximation algorithm for the SM-k-FLP.

Keywords

Approximation algorithm Facility location Local search 

Notes

Acknowledgements

The research of the first author is supported by Higher Educational Science and Technology Program of Shandong Province (No. J15LN22). The second author is supported by Natural Science Foundation of China (No. 11531014). The fourth author is supported by Natural Science Foundation of China (No. 61672323) and Natural Science Foundation of Shandong Province (ZR2016AM28). The fifth author is supported by Beijing Excellent Talents Funding (No. 2014000020124G046). A preliminary version of the paper appeared in Proceedings of the 11th Annual International Conference on Combinatorial Optimization and Applications, Shanghai, China, 2017.

References

  1. Arya V, Garg N, Khandekar R, Meyerson A, Munagala K, Pandit V (2004) Local search heuristics for \(k\)-median and facility location problems. SIAM J Comput 33:544–562MathSciNetCrossRefMATHGoogle Scholar
  2. Byrka J, Aardal KI (2010) An optimal bifactor approximation algorithm for the metric uncapacitated facility location problem. SIAM J Comput 39:2212–2231MathSciNetCrossRefMATHGoogle Scholar
  3. Byrka J, Pensyl T, Rybicki B, Srinivasan A (2014) An improved approximation for \(k\)-median, and positive correlation in budgeted optimization. In: Proceedings of SODA, pp 737–756Google Scholar
  4. Charikar M, Guha S (2005) Improved combinatorial algorithms for facility location problems. SIAM J Comput 34:803–824MathSciNetCrossRefMATHGoogle Scholar
  5. Charikar M, Guha S, Tardos E, Shmoys DB (2002) A constant-factor approximation algorithm for the \(k\)-median problem. J Comput Syst Sci 1:129–149MathSciNetCrossRefMATHGoogle Scholar
  6. Chen X, Chen B (1999) Approximation algorithms for soft-capacitated facility location in capacitated network design. Algorithmica 53:263–297MathSciNetCrossRefMATHGoogle Scholar
  7. Chudak FA, Shmoys DB (2003) Improved approximation algorithms for the uncapacitated facility location problem. SIAM J Comput 33:1–25MathSciNetCrossRefMATHGoogle Scholar
  8. Fernandes CG, Meira LA, Miyazawa FK, Pedrosa LL (2015) A systematic approach to bound factor-revealing LPs and its application to the metric and squared metric facility location problems. Math Program 153:655–685MathSciNetCrossRefMATHGoogle Scholar
  9. Guha S, Khuller S (1999) Greedy strikes back: improved facility location Algorithms. J Algorithms 31:228–248MathSciNetCrossRefMATHGoogle Scholar
  10. Jain K, Mahdian M, Markakis E, Saberi A, Vazirani V (2003) Greedy facility location algorithms analyzed using dual fitting with factor-revealing LP. J ACM 50:795–824MathSciNetCrossRefMATHGoogle Scholar
  11. Jain K, Vazirani VV (2001) Approximation algorithms for metric facility location and \(k\)-median problems using the primal-dual schema and Lagrangian relaxation. J ACM 48:274–296MathSciNetCrossRefMATHGoogle Scholar
  12. Kanungo T, Mount DM, Netanyahu NS, Piatko CD, Silverman R, Wu AY (2004) A local search approximation algorithm for \(k\)-means clustering. Comput Geom Theory Appl 2:89–112MathSciNetCrossRefMATHGoogle Scholar
  13. Li S (2013) A 1.488 approximation algorithm for the uncapacitated facility location problem. Inf Comput 222:45–58MathSciNetCrossRefMATHGoogle Scholar
  14. Li S, Svensson O (2016) Approximating \(k\)-median via pseudo-approximation. SIAM J Comput 45:530–547MathSciNetCrossRefMATHGoogle Scholar
  15. Mahdian M, Ye Y, Zhang J (2006) Approximation algorithms for metric facility location problems. SIAM J Comput 36:411–432MathSciNetCrossRefMATHGoogle Scholar
  16. Shmoys DB, Tardos E, Aardal K (1997) Approximation algorithms for facility location problems. In: Proceedings of SOTC, pp 265–274Google Scholar
  17. Zhang J, Chen B, Ye Y (2005) A multiexchange local search algorithm for the capacitated facility location problem. Math Oper Res 30:389–403MathSciNetCrossRefMATHGoogle Scholar
  18. Zhang P (2007) A new approximation algorithm for the \(k\)-facility location problem. Theor Comput Sci 384:126–135MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  • Dongmei Zhang
    • 1
  • Dachuan Xu
    • 2
  • Yishui Wang
    • 3
  • Peng Zhang
    • 4
  • Zhenning Zhang
    • 3
  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.Department of Information and Operations Research, College of Applied SciencesBeijing University of TechnologyBeijingPeople’s Republic of China
  4. 4.School of Computer Science and TechnologyShandong UniversityJinanPeople’s Republic of China

Personalised recommendations