Skip to main content

Smoothed Analysis of Local Search Algorithms

  • Conference paper
  • First Online:
  • 1730 Accesses

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9214))

Abstract

Smoothed analysis is a method for analyzing the performance of algorithms for which classical worst-case analysis fails to explain the performance observed in practice. Smoothed analysis has been applied to explain the performance of a variety of algorithms in the last years.

One particular class of algorithms where smoothed analysis has been used successfully are local search algorithms. We give a survey of smoothed analysis, in particular applied to local search algorithms.

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

Buying options

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Andoni, A., Krauthgamer, R.: The smoothed complexity of edit distance. ACM Transactions on Algorithms 8(4), 44:1–44:25 (2012)

    Google Scholar 

  2. Arthur, D., Manthey, B., Röglin, H.: Smoothed analysis of the \(k\)-means method. Journal of the ACM 58(5) (2011)

    Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  4. Banderier, C., Beier, R., Mehlhorn, K.: Smoothed analysis of three combinatorial problems. In: Rovan, B., Vojtáš, P. (eds.) MFCS 2003. LNCS, vol. 2747, pp. 198–207. Springer, Heidelberg (2003)

    Chapter  Google Scholar 

  5. Becchetti, L., Leonardi, S., Marchetti-Spaccamela, A., Schäfer, G., Vredeveld, T.: Average case and smoothed competitive analysis of the multilevel feedback algorithm. Mathematics of Operations Research 31(1), 85–108 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  6. Beier, R., Röglin, H., Vöcking, B.: The smoothed number of pareto optimal solutions in bicriteria integer optimization. In: Fischetti, M., Williamson, D.P. (eds.) IPCO 2007. LNCS, vol. 4513, pp. 53–67. Springer, Heidelberg (2007)

    Chapter  Google Scholar 

  7. Beier, R., Vöcking, B.: Random knapsack in expected polynomial time. Journal of Computer and System Sciences 69(3), 306–329 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  8. Beier, R., Vöcking, B.: Typical properties of winners and losers in discrete optimization. SIAM Journal on Computing 35(4), 855–881 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  9. de Berg, M., Haverkort, H.J., Tsirogiannis, C.P.: Visibility maps of realistic terrains have linear smoothed complexity. Journal of Computational Geometry 1(1), 57–71 (2010)

    MathSciNet  Google Scholar 

  10. Berger, A., Röglin, H., van der Zwaan, R.: Internet routing between autonomous systems: Fast algorithms for path trading. Discrete Applied Mathematics 185, 8–17 (2015)

    Article  MathSciNet  Google Scholar 

  11. Bläser, M., Manthey, B.: Smoothed complexity theory. ACM Transactions on Computation Theory (to appear)

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  13. Blum, A.L., Dunagan, J.D.: Smoothed analysis of the perceptron algorithm for linear programming. In: Proc. of the 13th Ann. ACM-SIAM Symp. on Discrete Algorithms (SODA), pp. 905–914. SIAM (2002)

    Google Scholar 

  14. Brunsch, T., Cornelissen, K., Manthey, B., Röglin, H.: Smoothed analysis of belief propagation for minimum-cost flow and matching. Journal of Graph Algorithms and Applications 17(6), 647–670 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  15. Brunsch, T., Cornelissen, K., Manthey, B., Röglin, H., Rösner, C.: Smoothed analysis of the successive shortest path algorithm. Computing Research Repository 1501.05493 [cs.DS], arXiv (2015), a preliminary version has been presented at SODA (2013)

    Google Scholar 

  16. Brunsch, T., Goyal, N., Rademacher, L., Röglin, H.: Lower bounds for the average and smoothed number of pareto-optima. Theory of Computing 10(10), 237–256 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  17. Brunsch, T., Röglin, H.: Improved smoothed analysis of multiobjective optimization. Journal of the ACM 62(1), 4 (2015)

    Article  MathSciNet  Google Scholar 

  18. Brunsch, T., Röglin, H., Rutten, C., Vredeveld, T.: Smoothed performance guarantees for local search. Mathematical Programming 146(1–2), 185–218 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  19. Bürgisser, P., Cucker, F., Lotz, M.: Smoothed analysis of complex conic condition numbers. Journal de Mathématiques Pures et Appliquées 86(4), 293–309 (2006)

    Article  MATH  Google Scholar 

  20. Bürgisser, P., Cucker, F., Lotz, M.: The probability that a slightly perturbed numerical analysis problem is difficult. Mathematics of Computation 77(263), 1559–1583 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  21. Chandra, B., Karloff, H., Tovey, C.: New results on the old \(k\)-opt algorithm for the traveling salesman problem. SIAM Journal on Computing 28(6), 1998–2029 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  22. Chaudhuri, S., Koltun, V.: Smoothed analysis of probabilistic roadmaps. Computational Geometry 42(8), 731–747 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  23. Chen, X., Deng, X., Teng, S.H.: Settling the complexity of computing two-player Nash equilibria. Journal of the ACM 56(3) (2009)

    Google Scholar 

  24. Coja-Oghlan, A., Feige, U., Frieze, A.M., Krivelevich, M., Vilenchik, D.: On smoothed \(k\)-CNF formulas and the Walksat algorithm. In: Proc. of the 20th Ann. ACM-SIAM Symp. on Discrete Algorithms (SODA), pp. 451–460. SIAM (2009)

    Google Scholar 

  25. Cornelissen, K., Manthey, B.: Smoothed analysis of the minimum-mean cycle canceling algorithm and the network simplex algorithm. In: Proc. of the 21st Ann. Int. Computing and Combinatorics Conf. (COCOON). LNCS. Springer (to appear, 2015)

    Google Scholar 

  26. Curticapean, R., Künnemann, M.: A quantization framework for smoothed analysis of euclidean optimization problems. In: Bodlaender, H.L., Italiano, G.F. (eds.) ESA 2013. LNCS, vol. 8125, pp. 349–360. Springer, Heidelberg (2013)

    Chapter  Google Scholar 

  27. Damerow, V., Manthey, B., Auf der Heide, F.M., Räcke, H., Scheideler, C., Sohler, C., Tantau, T.: Smoothed analysis of left-to-right maxima with applications. ACM Transactions on Algorithms 8(3) (2012)

    Google Scholar 

  28. Deshpande, A., Spielman, D.A.: Improved smoothed analysis of the shadow vertex simplex method. In: Proc. of the 46th Ann. IEEE Symp. on Foundations of Computer Science (FOCS), pp. 349–356. IEEE Computer Society (2005)

    Google Scholar 

  29. Dunagan, J., Spielman, D.A., Teng, S.H.: Smoothed analysis of condition numbers and complexity implications for linear programming. Mathematical Programming 126(2), 315–350 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  30. Elsässer, R., Tscheuschner, T.: Settling the complexity of local max-cut (almost) completely. In: Aceto, L., Henzinger, M., Sgall, J. (eds.) ICALP 2011, Part I. LNCS, vol. 6755, pp. 171–182. Springer, Heidelberg (2011)

    Chapter  Google Scholar 

  31. Englert, M., Röglin, H., Vöcking, B.: Worst case and probabilistic analysis of the 2-Opt algorithm for the TSP. Algorithmica 68(1), 190–264 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  32. Etscheid, M.: Performance guarantees for scheduling algorithms under perturbed machine speeds. Discrete Applied Mathematics (to appear)

    Google Scholar 

  33. Etscheid, M., Röglin, H.: Smoothed analysis of local search for the maximum-cut problem. In: Proc. of the 25th Ann. ACM-SIAM Symp. on Discrete Algorithms (SODA), pp. 882–889. SIAM (2014)

    Google Scholar 

  34. Feige, U.: Refuting smoothed 3CNF formulas. In: Proc. of the 48th Ann. IEEE Symp. on Foundations of Computer Science (FOCS), pp. 407–417. IEEE Computer Society (2007)

    Google Scholar 

  35. Flaxman, A.D., Frieze, A.M.: The diameter of randomly perturbed digraphs and some applications. Random Structures and Algorithms 30(4), 484–504 (2007)

    Article  MathSciNet  Google Scholar 

  36. 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)

    Article  MathSciNet  MATH  Google Scholar 

  37. Friedrich, T., Sauerwald, T., Vilenchik, D.: Smoothed analysis of balancing networks. Random Structures and Algorithms 39(1), 115–138 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  38. Kalai, A.T., Samorodnitsky, A., Teng, S.H.: Learning and smoothed analysis. In: Proc. of the 50th Ann. IEEE Symp. on Foundations of Computer Science (FOCS), pp. 395–404. IEEE Computer Society (2009)

    Google Scholar 

  39. Karger, D., Onak, K.: Polynomial approximation schemes for smoothed and random instances of multidimensional packing problems. In: Proc. of the 18th Ann. ACM-SIAM Symp. on Discrete Algorithms (SODA), pp. 1207–1216. SIAM (2007)

    Google Scholar 

  40. Kelner, J.A., Nikolova, E.: On the hardness and smoothed complexity of quasi-concave minimization. In: Proc. of the 48th Ann. IEEE Symp. on Foundations of Computer Science (FOCS), pp. 472–482 (2007)

    Google Scholar 

  41. Krivelevich, M., Sudakov, B., Tetali, P.: On smoothed analysis in dense graphs and formulas. Random Structures and Algorithms 29(2), 180–193 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  42. Künnemann, M., Manthey, B.: Towards understanding the smoothed approximation ratio of the 2-opt heuristic. In: Proc. of the 42nd Int. Coll. on Automata, Languages and Programming (ICALP). LNCS. Springer (to appear, 2015)

    Google Scholar 

  43. Lin, S., Kernighan, B.W.: An effective heuristic for the traveling-salesman problem. Operations Research 21(2), 498–516 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  44. Manthey, B., Plociennik, K.: Approximating independent set in perturbed graphs. Discrete Applied Mathematics 161(12), 1761–1768 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  45. Manthey, B., Reischuk, R.: Smoothed analysis of binary search trees. Theoretical Computer Science 378(3), 292–315 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  46. Manthey, B., Röglin, H.: Smoothed analysis: Analysis of algorithms beyond worst case. it - Information Technology 53(6), 280–286 (2011)

    Article  Google Scholar 

  47. Manthey, B., Röglin, H.: Worst-case and smoothed analysis of \(k\)-means clustering with Bregman divergences. Journal of Computational Geometry 4(1), 94–132 (2013)

    MathSciNet  Google Scholar 

  48. Manthey, B., Veenstra, R.: Smoothed analysis of the 2-Opt heuristic for the TSP: polynomial bounds for gaussian noise. In: Cai, L., Cheng, S.-W., Lam, T.-W. (eds.) ISAAC 2013. LNCS, vol. 8283, pp. 579–589. Springer, Heidelberg (2013)

    Chapter  Google Scholar 

  49. Moitra, A., O’Donnell, R.: Pareto optimal solutions for smoothed analysts. SIAM Journal on Computing 41(5), 1266–1284 (2012)

    Article  MathSciNet  Google Scholar 

  50. Röglin, H., Teng, S.H.: Smoothed analysis of multiobjective optimization. In: Proc. of the 50th Ann. IEEE Symp. on Foundations of Computer Science (FOCS), pp. 681–690. IEEE Computer Society (2009)

    Google Scholar 

  51. Röglin, H., Vöcking, B.: Smoothed analysis of integer programming. Mathematical Programming 110(1), 21–56 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  52. Sankar, A., Spielman, D.A., Teng, S.H.: Smoothed analysis of the condition numbers and growth factors of matrices. SIAM Journal on Matrix Analysis and Applications 28(2), 446–476 (2006)

    Article  MathSciNet  Google Scholar 

  53. Schäfer, G., Sivadasan, N.: Topology matters: Smoothed competitiveness of metrical task systems. Theoretical Computer Science 241(1–3), 216–246 (2005)

    Article  Google Scholar 

  54. Spielman, D.A., Teng, S.-H.: Smoothed analysis. In: Dehne, F., Sack, J.-R., Smid, M. (eds.) WADS 2003. LNCS, vol. 2748, pp. 256–270. Springer, Heidelberg (2003)

    Chapter  Google Scholar 

  55. Spielman, D.A., Teng, S.H.: Smoothed analysis of termination of linear programming algorithms. Mathematical Programming, Series B 97(1–2), 375–404 (2003)

    MathSciNet  MATH  Google Scholar 

  56. 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)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  58. Tao, T., Vu, V.H.: Smooth analysis of the condition number and the least singular value. Mathematics of Computation 79(272), 2333–2352 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  59. Vershynin, R.: Beyond Hirsch conjecture: Walks on random polytopes and smoothed complexity of the simplex method. SIAM Journal on Computing 39(2), 646–678 (2009)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Bodo Manthey .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer International Publishing Switzerland

About this paper

Cite this paper

Manthey, B. (2015). Smoothed Analysis of Local Search Algorithms. In: Dehne, F., Sack, JR., Stege, U. (eds) Algorithms and Data Structures. WADS 2015. Lecture Notes in Computer Science(), vol 9214. Springer, Cham. https://doi.org/10.1007/978-3-319-21840-3_43

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-21840-3_43

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-21839-7

  • Online ISBN: 978-3-319-21840-3

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics