A Quantization Framework for Smoothed Analysis of Euclidean Optimization Problems

  • Radu Curticapean
  • Marvin Künnemann
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8125)

Abstract

We consider the smoothed analysis of Euclidean optimization problems. Here, input points are sampled according to density functions that are bounded by a sufficiently small smoothness parameter φ. For such inputs, we provide a general and systematic approach that allows to design linear-time approximation algorithms whose output is asymptotically optimal, both in expectation and with high probability.

Applications of our framework include maximum matching, maximum TSP, and the classical problems of k-means clustering and bin packing. Apart from generalizing corresponding average-case analyses, our results extend and simplify a polynomial-time probable approximation scheme on multidimensional bin packing on φ-smooth instances, where φ is constant (Karger and Onak, SODA 2007).

Both techniques and applications of our rounding-based approach are orthogonal to the only other framework for smoothed analysis on Euclidean problems we are aware of (Bläser et al., Algorithmica 2012).

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Anstee, R.P.: A polynomial algorithm for b-matchings: An alternative approach. Information Processing Letters 24(3), 153–157 (1987)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Arthur, D., Manthey, B., Röglin, H.: Smoothed analysis of the k-means method. Journal of the ACM 58(5), 19:1–19:31 (2011)Google Scholar
  3. 3.
    Arthur, D., Vassilvitskii, S.: Worst-case and smoothed analysis of the ICP algorithm, with an application to the k-means method. SIAM Journal on Computing 39(2), 766–782 (2009)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Avis, D.: A survey of heuristics for the weighted matching problem. Networks 13(4), 475–493 (1983)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Awasthi, P., Blum, A., Sheffet, O.: Center-based clustering under perturbation stability. Information Processing Letters 112(1-2), 49–54 (2012)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Bansal, N., Correa, J.É.R., Kenyon, C., Sviridenko, M.: Bin packing in multiple dimensions: Inapproximability results and approximation schemes. Mathematics of Operations Research 31, 31–49 (2006)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Barvinok, A., Fekete, S.P., Johnson, D.S., Tamir, A., Woeginger, G.J., Woodroofe, R.: The geometric maximum traveling salesman problem. J. ACM 50(5), 641–664 (2003)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Barvinok, A.I.: Two algorithmic results for the traveling salesman problem. Mathematics of Operations Research 21(1), 65–84 (1996)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Beier, R., Vöcking, B.: Typical properties of winners and losers in discrete optimization. SIAM Journal on Computing 35(4), 855–881 (2006)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Bläser, M., Manthey, B., Raghavendra Rao, B.V.: Smoothed Analysis of Partitioning Algorithms for Euclidean Functionals. Algorithmica 66(2), 397–418 (2013)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Boros, E., Elbassioni, K., Fouz, M., Gurvich, V., Makino, K., Manthey, B.: Stochastic mean payoff games: Smoothed analysis and approximation schemes. In: Aceto, L., Henzinger, M., Sgall, J. (eds.) ICALP 2011, Part I. LNCS, vol. 6755, pp. 147–158. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  12. 12.
    Chen, K.: On coresets for k-median and k-means clustering in metric and euclidean spaces and their applications. SIAM Journal on Computing 39(3), 923–947 (2009)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Dasgupta, S.: The hardness of k-means clustering. Technical report cs2007-0890, University of California, San Diego (2007)Google Scholar
  14. 14.
    Dyer, M.E., Frieze, A.M., McDiarmid, C.J.H.: Partitioning heuristics for two geometric maximization problems. Operations Research Letters 3(5), 267–270 (1984)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Englert, M., Röglin, H., Vöcking, B.: Worst case and probabilistic analysis of the 2-opt algorithm for the TSP: Extended abstract. In: 18th Ann. ACM-SIAM Symp. on Discrete Algorithms, SODA 2007, pp. 1295–1304. SIAM (2007)Google Scholar
  16. 16.
    Fekete, S.P., Meijer, H., Rohe, A., Tietze, W.: Solving a ”hard” problem to approximate an ”easy” one: Heuristics for maximum matchings and maximum traveling salesman problems. ACM J. on Experimental Algorithmics. 7, 11 (2002)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Feldman, D., Monemizadeh, M., Sohler, C.: A PTAS for k-means clustering based on weak coresets. In: 23rd Ann. Symp. on Computational Geometry, SCG 2007, pp. 11–18. ACM (2007)Google Scholar
  18. 18.
    Fernandez de la Vega, W., Lueker, G.: Bin packing can be solved within 1 + ε in linear time. Combinatorica 1(4), 349–355 (1981)MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Gabow, H.N.: An efficient implementation of Edmonds’ algorithm for maximum matching on graphs. Journal of the ACM 23(2), 221–234 (1976)MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Har-Peled, S., Mazumdar, S.: On coresets for k-means and k-median clustering. In: 36th Ann. ACM Symp. on Theory of Computing, STOC 2004, pp. 291–300 (2004)Google Scholar
  21. 21.
    Inaba, M., Katoh, N., Imai, H.: Applications of weighted Voronoi diagrams and randomization to variance-based k-clustering (extended abstract). In: 10th Annual Symp. on Computational Geometry, SCG 1994, pp. 332–339 (1994)Google Scholar
  22. 22.
    Karger, D., Onak, K.: Polynomial approximation schemes for smoothed and random instances of multidimensional packing problems. In: 18th Ann. ACM-SIAM Symp. on Discrete Algorithms, SODA 2007, pp. 1207–1216 (2007)Google Scholar
  23. 23.
    Karp, R.M., Luby, M., Marchetti-Spaccamela, A.: A probabilistic analysis of multidimensional bin packing problems. In: 16th Annual ACM Symp. on Theory of Computing, STOC 1984, pp. 289–298. ACM, New York (1984)Google Scholar
  24. 24.
    Mahajan, M., Nimbhorkar, P., Varadarajan, K.: The planar k-means problem is NP-hard. Theoretical Computer Science 442, 13–21 (2012)MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    Plotkin, S.A., Shmoys, D.B., Tardos, É.: Fast approximation algorithms for fractional packing and covering problems. Mathematics of Operations Research 20(2), 257 (1995)MathSciNetMATHCrossRefGoogle Scholar
  26. 26.
    Spielman, D.A., Teng, S.-H.: Smoothed analysis of algorithms: Why the simplex algorithm usually takes polynomial time. Journal of the ACM 51(3), 385–463 (2004)MathSciNetMATHCrossRefGoogle Scholar
  27. 27.
    Steele, J.M.: Subadditive Euclidean functionals and nonlinear growth in geometric probability. The Annals of Probability 9(3), 365–376 (1981)MathSciNetMATHCrossRefGoogle Scholar
  28. 28.
    Weber, M., Liebling, T.M.: Euclidean matching problems and the metropolis algorithm. Mathematical Methods of Operations Research 30(3), A85–A110 (1986)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Radu Curticapean
    • 1
  • Marvin Künnemann
    • 1
    • 2
  1. 1.Universität des SaarlandesSaarbrückenGermany
  2. 2.Max-Planck-Institut für InformatikSaarbrückenGermany

Personalised recommendations