Advertisement

Abstract

Smoothed analysis is a new way of analyzing algorithms introduced by Spielman and Teng (J. ACM, 2004). Classical methods like worst-case or average-case analysis have accompanying complexity classes, like P and AvgP, respectively. While worst-case or average-case analysis give us a means to talk about the running time of a particular algorithm, complexity classes allows us to talk about the inherent difficulty of problems.

Smoothed analysis is a hybrid of worst-case and average-case analysis and compensates some of their drawbacks. Despite its success for the analysis of single algorithms and problems, there is no embedding of smoothed analysis into computational complexity theory, which is necessary to classify problems according to their intrinsic difficulty.

We propose a framework for smoothed complexity theory, define the relevant classes, and prove some first results.

Keywords

Compression Function Heuristic Scheme Perturbation Model Disjoint Support Computational Complexity Theory 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Arthur, D., Manthey, B., Röglin, H.: Smoothed analysis of the k-means method. J. ACM 58(5) (2011)Google Scholar
  2. 2.
    Beier, R., Vöcking, B.: Random knapsack in expected polynomial time. J. Comput. System Sci. 69(3), 306–329 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Beier, R., Vöcking, B.: Typical properties of winners and losers in discrete optimization. SIAM J. Comput. 35(4), 855–881 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Ben-David, S., Chor, B., Goldreich, O., Luby, M.: On the theory of average case complexity. J. Comput. System Sci. 44(2), 193–219 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Bläser, M., Manthey, B., Rao, B.V.R.: Smoothed analysis of partitioning algorithms for Euclidean functionals. Algorithmica (to appear)Google Scholar
  6. 6.
    Blum, A.L., Spencer, J.: Coloring random and semi-random k-colorable graphs. J. Algorithms 19(2), 204–234 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Bogdanov, A., Trevisan, L.: Average-case complexity. Foundations and Trends in Theoret. Comput. Sci. 2(1), 1–106 (2006)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Bohman, T., Frieze, A.M., Krivelevich, M., Martin, R.: Adding random edges to dense graphs. Random Struct. Algorithms 24(2), 105–117 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Coja-Oghlan, A.: Colouring semirandom graphs. Combin. Probab. Comput. 16(4), 515–552 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Coja-Oghlan, A.: Solving NP-hard semirandom graph problems in polynomial expected time. J. Algorithms 62(1), 19–46 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Coja-Oghlan, A., Feige, U., Frieze, A.M., Krivelevich, M., Vilenchik, D.: On smoothed k-CNF formulas and the Walksat algorithm. In: Proc. 20th Ann. Symp. on Discrete Algorithms (SODA), pp. 451–460. SIAM (2009)Google Scholar
  12. 12.
    Damerow, V., Manthey, B., Meyer auf der Heide, F., Räcke, H., Scheideler, C., Sohler, C., Tantau, T.: Smoothed analysis of left-to-right maxima with applications. ACM Trans. Algorithms 8(3), article 30Google Scholar
  13. 13.
    Englert, M., Röglin, H., Vöcking, B.: Worst case and probabilistic analysis of the 2-Opt algorithm for the TSP. In: Proc. 18th Ann. Symp. on Discrete Algorithms (SODA), pp. 1295–1304. SIAM (2007)Google Scholar
  14. 14.
    Feige, U.: Refuting smoothed 3CNF formulas. In: Proc. 48th Ann. Symp. on Foundations of Computer Science (FOCS), pp. 407–417. IEEE (2007)Google Scholar
  15. 15.
    Feige, U., Kilian, J.: Heuristics for semirandom graph problems. J. Comput. System Sci. 63(4), 639–671 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Fouz, M., Kufleitner, M., Manthey, B., Zeini Jahromi, N.: On smoothed analysis of quicksort and Hoare’s find. Algorithmica 62(3-4), 879–905 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Garey, M.R., Johnson, D.S.: Computers and Intractability. W. H. Freeman and Company (1979)Google Scholar
  18. 18.
    Gurevich, Y.: Average case completeness. J. Comput. System Sci. 42(3), 346–398 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Krivelevich, M., Sudakov, B., Tetali, P.: On smoothed analysis in dense graphs and formulas. Random Struct. Algorithms 29(2), 180–193 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Levin, L.A.: Average case complete problems. SIAM J. Comput. 15(1), 285–286 (1986)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Li, M., Vitányi, P.M.B.: Average case complexity under the universal distribution equals worst-case complexity. Inform. Process. Lett. 42(3), 145–149 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Manthey, B., Röglin, H.: Smoothed analysis: Analysis of algorithms beyond worst case. it – Information Technology 53(6) (2011)Google Scholar
  23. 23.
    Moitra, A., O’Donnell, R.: Pareto optimal solutions for smoothed analysts. In: Proc. 43rd Ann. Symp. on Theory of Computing (STOC), pp. 225–234. ACM (2011)Google Scholar
  24. 24.
    Röglin, H., Teng, S.-H.: Smoothed analysis of multiobjective optimization. In: Proc. 50th Ann. Symp. on Foundations of Computer Science (FOCS), pp. 681–690. IEEE (2009)Google Scholar
  25. 25.
    Röglin, H., Vöcking, B.: Smoothed analysis of integer programming. Math. Prog. 110(1), 21–56 (2007)zbMATHCrossRefGoogle Scholar
  26. 26.
    Spielman, D.A., Teng, S.-H.: Smoothed analysis of termination of linear programming algorithms. Math. Prog. 97(1-2), 375–404 (2003)MathSciNetzbMATHGoogle Scholar
  27. 27.
    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)MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Spielman, D.A., Teng, S.-H.: Smoothed analysis: An attempt to explain the behavior of algorithms in practice. Commun. ACM 52(10), 76–84 (2009)CrossRefGoogle Scholar
  29. 29.
    Vershynin, R.: Beyond Hirsch conjecture: Walks on random polytopes and smoothed complexity of the simplex method. SIAM J. Comput. 39(2), 646–678 (2009)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Markus Bläser
    • 1
  • Bodo Manthey
    • 2
  1. 1.Saarland UniversityGermany
  2. 2.University of TwenteThe Netherlands

Personalised recommendations