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Efficient Approximation and Online Algorithms pp 292–320Cite as

Approximation Algorithms for the k-Median Problem

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 3484))

Abstract

The k-median problem is a central problem in Operations Research that has captured the attention of the Algorithms community in recent years. Despite its importance, a non-trivial approximation algorithm for the problem eluded researchers for a long time. Remarkably, a succession of papers with ever improved performance ratios have been written in the last couple of years. We review some of the approaches that have been used to design approximation algorithms for this problem, and we also present some of the known results about the hardness of approximating the optimum solution for the k-median problem.

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References

  1. Archer, A., Rajagopalan, R., Shmoys, D.: Lagrangian relaxation for the k-median problem: new insights and continuity properties. In: Di Battista, G., Zwick, U. (eds.) ESA 2003. LNCS, vol. 2832, pp. 31–42. Springer, Heidelberg (2003)

    Chapter  Google Scholar 

  2. Ahn, S., Cooper, A., Cornuejols, G., Frieze, A.: Probabilistic analysis of a relaxation for the k-median problem. Mathematics of Operations Research 13, 1–31 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  3. Andrews, M., Zhang, L.: The access network design problem. In: Proceedings of the 39th Annual IEEE Symposium on Foundations of Computer Science, pp. 40–59 (1998)

    Google Scholar 

  4. Arora, S., Raghavan, P., Rao, S.: Approximation schemes for Euclidean k-medians and related problems. In: Proceedings of the 30th Annual ACM Symposium on Theory of Computing, pp. 106–113 (1998)

    Google Scholar 

  5. Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k-median and facility location problems. In: Proceedings of the ACM Symposium on Theory of Computing, pp. 21–29 (2001)

    Google Scholar 

  6. Balinski, M.L.: On finding integer solutions to linear programs. In: Proceedings of the IBM Scientific Computing Symposium on Combinatorial Problems, pp. 225–248 (1966)

    Google Scholar 

  7. Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: Proceedings of the 37th Annual IEEE Symposium on Foundations of Computer Science, pp. 184–193 (1996)

    Google Scholar 

  8. Bartal, Y.: On approximating arbitrary metrics by tree metrics. In: Proceedings of the 30th Annual ACM Symposium on Theory of Computing, pp. 161–168 (1998)

    Google Scholar 

  9. Bradley, P.S., Fayad, U.M., Mangasarian, O.L.: Mathematical programming for data mining: formulations and challenges, Microsoft Technical Report (1998)

    Google Scholar 

  10. Bumb, A.F., Kern, W.: A simple dual ascent algorithm for the multilevel facility location problem. In: Proceedings of RANDOM-APPROX, pp. 55–62 (2001)

    Google Scholar 

  11. Charikar, M., Chekuri, C., Goel, A., Guha, S.: Rounding via trees: deterministic approximation algorithms for group Steiner trees and k-median. In: Proceedings of the 30th Annual ACM Symposium on Theory of Computing, pp. 114–123 (1998)

    Google Scholar 

  12. Charikar, M., Guha, S.: Improved combinatorial algorithms for the facility location and k-median problems. In: Proceedings of the 40th Annual Symposium on Foundations of Computer Science, pp. 378–388 (1999)

    Google Scholar 

  13. Charikar, M., Guha, S., Tardos, E., Shmoys, D.B.: A constant-factor approximation algorithm for the k-median problem. In: Proceedings of the Thirty-First Symposium on Theory of Computing, pp. 1–10 (1999)

    Google Scholar 

  14. Cheriyan, J., Ravi, R.: Approximation algorithms for network problems, Lecture Notes, University of Waterloo (1998)

    Google Scholar 

  15. Chudak, F., Williamson, D.: Improved approximation algorithms for capacitated facility location problems. In: Proceedings of the 7th Conference on Integer Programming and Combinatorial Optimization (1999)

    Google Scholar 

  16. Chvátal, V.: A greedy heuristic for the set covering problem. Mathematics of Operations Research 4, 233–235 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  17. Chvátal, V.: Linear Programming. W.H. Freeman and Company, New York (1983)

    MATH  Google Scholar 

  18. Cornuejols, G., Nemhauser, G.L., Wolsey, L.A.: The uncapacitated facility location problem. In: Mirchandani, P., Francis, R. (eds.) Discrete Location Theory. John Wiley and Sons, New York (1990)

    Google Scholar 

  19. Guha, S., Meyerson, A., Munagala, K.: Hierarchical placement and network design problems. In: Proceedings of the 14th Annual IEEE Symposium on Foundations of Computer Science (2000)

    Google Scholar 

  20. Guha, S., Meyerson, A., Munagala, K.: Improved combinatorial algorithms for single sink edge installation problems, Technical Report STAN-CS-TN00-96, Stanford University (2000)

    Google Scholar 

  21. Guha, S., Meyerson, A., Munagala, K.: Improved algorithms for fault tolerant facility location. In: Proceedings of the ACM-SIAM Symposium on Discrete Algorithms, pp. 636–641 (2001)

    Google Scholar 

  22. Jamin, S., Jin, C., Jin, Y., Raz, D., Shavitt, Y., Zhang, L.: On the placement of internet instrumentations. In: Proceedings of IEEE INFOCOM, pp. 26–30 (2000)

    Google Scholar 

  23. Jain, K., Vazirani, V.V.: Approximation algorithms for metric facility location and k-median problems using the primal-dual schema and Lagrangian relaxation. In: Proceedings of the 40th Annual Symposium on Foundations of Computer Science (1999); Also Journal of the ACM 48, 274–296 (2001)

    Google Scholar 

  24. Johnson, D.S.: Approximation algorithms for combinatorial problems. Journal of Computer and System Sciences 9, 256–278 (1974)

    Article  MATH  MathSciNet  Google Scholar 

  25. Kariv, O., Hakimi, S.L.: An algorithmic approach to network location problems, Part II: p-medians. SIAM Journal on Applied Mathematics, 539–560 (1979)

    Google Scholar 

  26. Kariv, O., Hakimi, S.L.: An algorithmic approach to network location problems. II: The p-medians. SIAM Journal on Applied Mathematics, 539–560 (1979)

    Google Scholar 

  27. Khachiyan, L.: A polynomial algorithm for linear programming. Doklady Akad. Nauk USSR 244(5), 1093–1096 (1979)

    MATH  MathSciNet  Google Scholar 

  28. Khuller, S., Pless, R., Sussman, Y.J.: Fault tolerant k-center problems. Theoretical Computer Science 242, 237–245 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  29. Kuehn, A.A., Hamburger, M.J.: A heuristic program for locating warehouses. Management Science 9, 643–666 (1963)

    Article  Google Scholar 

  30. Korupolu, M.R., Plaxton, C.G., Rajaraman, R.: Analysis of a local search heuristic for facility location problems. In: Proceedings of the 9th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 1–10 (1998); Also in Journal of Algorithms 37, 146–188 (2000)

    Google Scholar 

  31. Krysta, P., Solis-Oba, R.: Approximation algorithms for bounded facility location. Journal of Combinatorial Optimization 5, 2–16 (2001)

    Article  MathSciNet  Google Scholar 

  32. Li, B., Golin, M., Italiano, G., Deng, X., Sohraby, K.: On the optimal placement of web proxies in the internet. In: Proceedings of IEEE INFOCOM, pp. 1282–1290 (1999)

    Google Scholar 

  33. Lin, J., Vitter, J.S.: ε-Approximations with minimum packing constraint violation. In: Proceedings of the 24th Annual ACM Symposium on Theory of Computing, pp. 771–782 (1992)

    Google Scholar 

  34. Lin, J., Vitter, J.S.: Approximation algorithms for geometric median problems. Information Processing Letters 44, 245–249 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  35. Lovász, L.: On the ratio of optimal integral and fractional covers. Discrete Mathematics 13, 383–390 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  36. Megiddo, N., Supowit, K.J.: On the complexity of some common geometric location problems. SIAM Journal on Computing 13, 182–196 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  37. Mahdian, M., Pál, M.: Universal facility location. In: Di Battista, G., Zwick, U. (eds.) ESA 2003. LNCS, vol. 2832, pp. 409–421. Springer, Heidelberg (2003)

    Chapter  Google Scholar 

  38. Mahdian, M., Ye, Y., Zhang, J.: Improved approximation algorithms for metric facility location problems. In: Jansen, K., Leonardi, S., Vazirani, V.V. (eds.) APPROX 2002. LNCS, vol. 2462, pp. 229–242. Springer, Heidelberg (2002)

    Chapter  Google Scholar 

  39. Mettu, R.R., Plaxton, C.G.: The online median problem. SIAM Journal on Computing 32, 816–832 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  40. Mulvey, J.M., Crowder, H.L.: Cluster Analysis: an application of Lagrangian relaxation. Management Science 25, 329–340 (1979)

    Article  MATH  Google Scholar 

  41. Murty, K.: Linear Programming. John Wiley & Sons, Chichester (1983)

    MATH  Google Scholar 

  42. Nemhauser, G.L., Wolsey, L.A.: Integer and Combinatorial Optimization. John Wiley and Sons, New York (1990)

    Google Scholar 

  43. Papadimitriou, C.H.: Worst case and probabilistic analysis of a geometric location problem. SIAM Journal on Computing 10(3), 542–557 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  44. Qiu, L., Padmanabhan, V.N., Voelker, G.: On the placement of web server replicas. In: Proceedings of IEEE INFOCOM (2001)

    Google Scholar 

  45. Raz, R., Safra, S.: A sub-constant error-probability low-degree test, and sub-constant error-probability PCP characterization of NP. In: Proceedings of the 29th Annual ACM Symposium on Theory of Computing, pp. 475–484 (1997)

    Google Scholar 

  46. Schrijver, A.: Theory of Linear and Integer Programming. John Wiley & Sons, Chichester (1986)

    MATH  Google Scholar 

  47. Tamir, A.: An O(pn 2) algorithm for the p-median and related problems on tree graphs. Operations Research Letters 19, 59–94 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  48. Ye, Y.: An O(n 3 L) potential reduction algorithm for linear programming. Mathematical Programming 50, 239–258 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  49. Zhang, J.: Approximating the two-level facility location problem via a quasi-greedy approach. In: Proceedings of the 15th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 801–810 (2004)

    Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Computer Science, The University of Western Ontario, London, Ontario, Canada

    Roberto Solis-Oba

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  1. Roberto Solis-Oba
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Editors and Affiliations

  1. IBISC, CNRS FRE 3190, Université d’Evry Val d’Essonne, Boulevard François Mitterand, 91025, Evry Cedex, France

    Evripidis Bampis

  2. Institute for Computer Science, University of Kiel, Olshausenstrasse 40, 24118, Kiel, Germany

    Klaus Jansen

  3. Brown University, 02918, Providence, RI, USA

    Claire Kenyon

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© 2006 Springer-Verlag Berlin Heidelberg

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Solis-Oba, R. (2006). Approximation Algorithms for the k-Median Problem. In: Bampis, E., Jansen, K., Kenyon, C. (eds) Efficient Approximation and Online Algorithms. Lecture Notes in Computer Science, vol 3484. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11671541_10

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  • DOI: https://doi.org/10.1007/11671541_10

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-32212-2

  • Online ISBN: 978-3-540-32213-9

  • eBook Packages: Computer ScienceComputer Science (R0)

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