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New Approximability Results for the Robust k-Median Problem

  • Sayan Bhattacharya
  • Parinya Chalermsook
  • Kurt Mehlhorn
  • Adrian Neumann
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8503)

Abstract

We consider a variant of the classical k-median problem, introduced by Anthony et al.[1]. In the Robust k-Median problem, we are given an n-vertex metric space (V,d) and m client sets \(\left\{ S_i \subseteq V \right\}_{i=1}^m\). We want to open a set F ⊆ V of k facilities such that the worst case connection cost over all client sets is minimized; that is, minimize \(\max_{i}\sum_{v \in S_i} d(F,v)\). Anthony et al. showed an O(logm) approximation algorithm for any metric and APX-hardness even in the case of uniform metric. In this paper, we show that their algorithm is nearly tight by providing Ω(logm/ loglogm) approximation hardness, unless \({\sf NP} \subseteq \bigcap_{\delta >0} {\sf DTIME}(2^{n^{\delta}})\). This result holds even for uniform and line metrics. To our knowledge, this is one of the rare cases in which a problem on a line metric is hard to approximate to within logarithmic factor. We complement the hardness result by an experimental evaluation of different heuristics that shows that very simple heuristics achieve good approximations for realistic classes of instances.

Keywords

Approximation Algorithm Hardness Result Line Metrics Approximability Result Polynomial Time Approximation Scheme 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Anthony, B.M., Goyal, V., Gupta, A., Nagarajan, V.: A plant location guide for the unsure: Approximation algorithms for min-max location problems. Math. Oper. Res. 35(1), 79–101 (2010) (Also in SODA 2008)Google Scholar
  2. 2.
    Arora, S.: Polynomial time approximation schemes for euclidean traveling salesman and other geometric problems. J. ACM 45(5), 753–782 (1998)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Arora, S., Lund, C., Motwani, R., Sudan, M., Szegedy, M.: Proof verification and the hardness of approximation problems. J. ACM 45(3), 501–555 (1998)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    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(3), 544–562 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Bansal, N., Khandekar, R., Könemann, J., Nagarajan, V., Peis, B.: On generalizations of network design problems with degree bounds. Math. Program. 141(1-2), 479–506 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Charikar, M., Guha, S.: Improved combinatorial algorithms for the facility location and k-median problems. In: FOCS, pp. 378–388. IEEE Computer Society (1999)Google Scholar
  7. 7.
    Charikar, M., Guha, S., Tardos, É., Shmoys, D.B.: A constant-factor approximation algorithm for the k-median problem. J. Comput. Syst. Sci. 65(1), 129–149 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Feige, U.: A threshold of ln n for approximating set cover. J. ACM 45(4), 634–652 (1998)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Hagerup, T., Rüb, C.: A guided tour of Chernoff bounds. Information Processing Letters 33(6), 305–308 (1990), http://www.sciencedirect.com/science/article/pii/002001909090214I CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Kolliopoulos, S.G., Rao, S.: A nearly linear-time approximation scheme for the euclidean k-median problem. In: Nešetřil, J. (ed.) ESA 1999. LNCS, vol. 1643, pp. 378–389. Springer, Heidelberg (1999)Google Scholar
  11. 11.
    Li, S., Svensson, O.: Approximating k-median via pseudo-approximation. In: Boneh, D., Roughgarden, T., Feigenbaum, J. (eds.) STOC, pp. 901–910. ACM (2013)Google Scholar
  12. 12.
    Lin, J.H., Vitter, J.S.: Approximation algorithms for geometric median problems. Inf. Process. Lett. 44(5), 245–249 (1992)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Lund, C., Yannakakis, M.: On the hardness of approximating minimization problems. J. ACM 41(5), 960–981 (1994)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Raz, R.: A parallel repetition theorem. SIAM J. Comput. 27(3), 763–803 (1998)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Sayan Bhattacharya
    • 1
  • Parinya Chalermsook
    • 1
  • Kurt Mehlhorn
    • 1
  • Adrian Neumann
    • 1
  1. 1.Max-Planck Institut für InformatikGermany

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