Skip to main content
SpringerLink
Log in

A Quantization Framework for Smoothed Analysis of Euclidean Optimization Problems

  • Published:
Algorithmica Aims and scope Submit manuscript

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 \(\phi \). For such inputs, we provide a general and systematic approach that allows designing 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 \(\phi \)-smooth instances, where \(\phi \) is constant (Karger and Onak in Polynomial approximation schemes for smoothed and random instances of multidimensional packing problems, pp 1207–1216, 2007). Both techniques and applications of our rounding-based approach are orthogonal to the only other framework for smoothed analysis of Euclidean problems we are aware of (Bläser et al. in Algorithmica 66(2):397–418, 2013).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. If the framework algorithm fails with probability at most p, then an o(1 / p)-approximation algorithm would also suffice to ensure expected asymptotic optimality. At this point, we require O(1)-approximations only for simplicity of presentation. In Sect. 6, we will make use of a slightly more precise analysis of the failure probability of the framework algorithm to use an n-approximation for bin packing.

References

  1. Anstee, R.P.: A polynomial algorithm for b-matchings: an alternative approach. Inf. Proc. Lett. 24(3), 153–157 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  2. Arthur, D., Manthey, B., Röglin, H.: Smoothed analysis of the k-means method. J. ACM 58(5), 19:1–19:31 (2011)

    Article  Google Scholar 

  3. Arthur, D., Vassilvitskii, S.: k-means++: the advantages of careful seeding. In: 18th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA’07, pp. 1027–1035. SIAM (2007)

  4. Arthur, D., Vassilvitskii, S.: Worst-case and smoothed analysis of the ICP algorithm, with an application to the k-means method. SIAM J. Comput. 39(2), 766–782 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  5. Avis, D.: A survey of heuristics for the weighted matching problem. Networks 13(4), 475–493 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  6. Awasthi, P., Blum, A., Sheffet, O.: Center-based clustering under perturbation stability. Inf. Proc. Lett. 112(1–2), 49–54 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  7. Bansal, N., Correa, J.É.R., Kenyon, C., Sviridenko, M.: Bin packing in multiple dimensions: inapproximability results and approximation schemes. Math. Oper. Res. 31, 31–49 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  8. 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)

    Article  MathSciNet  Google Scholar 

  9. Barvinok, A.I.: Two algorithmic results for the traveling salesman problem. Math. Oper. Res. 21(1), 65–84 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  10. Beier, R., Vöcking, B.: Typical properties of winners and losers in discrete optimization. SIAM J. Comput. 35(4), 855–881 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  11. Bern, M., Eppstein, D.: Worst-case bounds for subadditive geometric graphs. In: 9th Annual Symposium on Computational Geometry, SCG’93, pp. 183–188, New York, NY, USA, ACM (1993)

  12. Bläser, M., Manthey, B., Rao, B.V.R.: Smoothed analysis of partitioning algorithms for Euclidean functionals. Algorithmica 66(2), 397–418 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  13. Boros, E., Elbassioni, K., Fouz, M., Gurvich, V., Makino, K., Manthey, B.: Stochastic mean payoff games: smoothed analysis and approximation schemes. In: 38th International Colloquium on Automata, Languages and Programming, ICALP’11, pp. 147–158. Springer (2011)

  14. Chen, K.: On coresets for k-median and k-means clustering in metric and Euclidean spaces and their applications. SIAM J. Comput. 39(3), 923–947 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  15. Curticapean, R., Künnemann, M.: A quantization framework for smoothed analysis of Euclidean optimization problems. In: 21st European Symposium on Algorithms, ESA’13, pp. 349–360. Springer, Berlin (2013)

  16. Dasgupta, S.: The hardness of k-means clustering. Technical report cs2007-0890, University of California, San Diego (2007)

  17. Duan, R., Pettie, S.: Approximating maximum weight matching in near-linear time. In: 51st Annual IEEE Symposium on Foundations of Computer Science, FOCS’10, pp. 673–682, Washington, DC, USA. IEEE Computer Society (2010)

  18. Dyer, M.E., Frieze, A.M., McDiarmid, C.J.H.: Partitioning heuristics for two geometric maximization problems. Oper. Res. Lett. 3(5), 267–270 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  19. Englert, M., Röglin, H., Vöcking, B.: Worst case and probabilistic analysis of the 2-opt algorithm for the TSP: extended abstract. In: 18th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA’07, pp. 1295–1304. SIAM (2007)

  20. 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. Exp. Algorithmics 7, 11 (2002)

    Article  MathSciNet  Google Scholar 

  21. Feldman, D., Monemizadeh, M., Sohler, C.: A PTAS for k-means clustering based on weak coresets. In: 23rd Annual Symposium on Computational Geometry, SCG’07, pp. 11–18. ACM (2007)

  22. Fernandez de la Vega, W., Lueker, G.: Bin packing can be solved within \(1 + \epsilon \) in linear time. Combinatorica 1(4), 349–355 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  23. Gabow, H.N.: An efficient implementation of Edmonds’ algorithm for maximum matching on graphs. J. ACM 23(2), 221–234 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  24. Har-Peled, S., Mazumdar, S.: On coresets for k-means and k-median clustering. In: 36th Annual ACM Symposium on Theory of Computing, STOC’04, pp. 291–300 (2004)

  25. Inaba, M., Katoh, N., Imai, H.: Applications of weighted Voronoi diagrams and randomization to variance-based k-clustering: (extended abstract). In: 10th Annual Symposium on Computational Geometry, SCG’94, pp. 332–339 (1994)

  26. Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k-means clustering. Comput. Geom. Theo. Appl. 28(2–3), 89–112 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  27. Karger, D., Onak, K.: Polynomial approximation schemes for smoothed and random instances of multidimensional packing problems. In: 18th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA’07, pp. 1207–1216 (2007)

  28. Karp, R.M., Luby, M., Marchetti-Spaccamela, A.: A probabilistic analysis of multidimensional bin packing problems. In: 16th Annual ACM Symposium on Theory of Computing, STOC’84, pp. 289–298, New York, NY, USA, ACM (1984)

  29. Mahajan, M., Nimbhorkar, P., Varadarajan, K.: The planar k-means problem is NP-hard. Theor. Comput. Sci. 442, 13–21 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  30. Manthey, B., Röglin, H.: Smoothed analysis: analysis of algorithms beyond worst case. Inf. Technol. 53(6), 280–286 (2011)

    Google Scholar 

  31. McDiarmid, C.: Concentration. In: Habib, M., McDiarmid, C., Ramirez-Alfonsin, J., Reed, B. (eds.) Probabilistic Methods for Algorithmic Discrete Mathematics, Volume 16 of Algorithms and Combinatorics, pp. 195–248. Springer, Berlin (1998)

    Chapter  Google Scholar 

  32. Plotkin, S.A., Shmoys, D.B., Tardos, É.: Fast approximation algorithms for fractional packing and covering problems. Math. Oper. Res. 20(2), 257 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  33. Spielman, D.A., Teng, S.: Smoothed analysis: an attempt to explain the behavior of algorithms in practice. Commun. ACM 52(10), 76–84 (2009)

    Article  Google Scholar 

  34. Spielman, D.A., Teng, S.-H.: Smoothed analysis of algorithms: why the simplex algorithm usually takes polynomial time. J. ACM 51(3), 385–463 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  35. Steele, J.M.: Subadditive Euclidean functionals and nonlinear growth in geometric probability. Ann. Probab. 9(3), 365–376 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  36. Weber, M., Liebling, T.M.: Euclidean matching problems and the metropolis algorithm. Math. Methods Oper. Res. 30(3), A85–A110 (1986)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgments

The authors are grateful to Markus Bläser for kindling their interest in smoothed analysis and for stimulating discussions, and to the anonymous reviewers of this article for providing helpful remarks.

Author information

Authors and Affiliations

  1. Saarbrücken Graduate School of Computer Science, Saarbrücken, Germany

    Radu Curticapean & Marvin Künnemann

  2. Department of Computer Science, Saarland University, Campus E1 3, 66123, Saarbrücken, Germany

    Radu Curticapean

  3. Max Planck Institute for Informatics, Campus E1 4, 66123, Saarbrücken, Germany

    Marvin Künnemann

Authors
  1. Radu Curticapean
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Marvin Künnemann
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Marvin Künnemann.

Additional information

An extended abstract of this article appeared in the Proceedings of the 21st European Symposium on Algorithms (ESA’13) [15].

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Curticapean, R., Künnemann, M. A Quantization Framework for Smoothed Analysis of Euclidean Optimization Problems. Algorithmica 73, 483–510 (2015). https://doi.org/10.1007/s00453-015-0043-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00453-015-0043-5

Keywords

Search

Navigation