Smoothed Analysis of Partitioning Algorithms for Euclidean Functionals

  • Markus Bläser
  • Bodo Manthey
  • B. V. Raghavendra Rao
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6844)


Euclidean optimization problems such as TSP and minimum-length matching admit fast partitioning algorithms that compute near-optimal solutions on typical instances.

We develop a general framework for the application of smoothed analysis to partitioning algorithms for Euclidean optimization problems. Our framework can be used to analyze both the running-time and the approximation ratio of such algorithms. We apply our framework to obtain smoothed analyses of Dyer and Frieze’s partitioning algorithm for Euclidean matching, Karp’s partitioning scheme for the TSP, a heuristic for Steiner trees, and a heuristic for degree-bounded minimum-length spanning trees.


Approximation Ratio Travel Salesman Problem Steiner Tree Steiner Tree Problem Partitioning Algorithm 
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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  1. 1.
    Ahuja, R.K., Magnanti, T.L., Orlin, J.B.: Network Flows. Prentice-Hall, Englewood Cliffs (1993)zbMATHGoogle Scholar
  2. 2.
    Anthes, B., Rüschendorf, L.: On the weighted Euclidean matching problem in R d dimensions. Applicationes Mathematicae 28(2), 181–190 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Arthur, D., Manthey, B., Röglin, H.: k-means has polynomial smoothed complexity. In: Proc. 50th Ann. IEEE Symp. on Foundations of Computer Science (FOCS), pp. 405–414. IEEE, Los Alamitos (2009)Google Scholar
  4. 4.
    Beier, R., Vöcking, B.: Random knapsack in expected polynomial time. J. Comput. System Sci. 69(3), 306–329 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Damerow, V., Sohler, C.: Extreme points under random noise. In: Albers, S., Radzik, T. (eds.) ESA 2004. LNCS, vol. 3221, pp. 264–274. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  6. 6.
    Dreyfus, S.E., Wagner, R.A.: The Steiner problem in graphs. Networks 1(3), 195–207 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Dyer, M.E., Frieze, A.M.: A partitioning algorithm for minimum weighted euclidean matching. Inform. Process. Lett. 18(2), 59–62 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Engels, C., Manthey, B.: Average-case approximation ratio of the 2-opt algorithm for the TSP. Oper. Res. Lett. 37(2), 83–84 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Englert, M., Röglin, H., Vöcking, B.: Worst case and probabilistic analysis of the 2-Opt algorithm for the TSP. In: Proc. 18th Ann. ACM-SIAM Symp. on Discrete Algorithms (SODA), pp. 1295–1304. SIAM, Philadelphia (2007)Google Scholar
  10. 10.
    Frieze, A.M., Yukich, J.E.: Probabilistic analysis of the traveling salesman problem. In: Gutin, G., Punnen, A.P. (eds.) The Traveling Salesman Problem and Its Variations, ch.7, pp. 257–308. Kluwer, Dordrecht (2002)Google Scholar
  11. 11.
    Garey, M.R., Graham, R.L., Johnson, D.S.: The complexity of computing Steiner minimal trees. SIAM J. Appl. Math. 32(4), 835–859 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Golin, M.J.: Limit theorems for minimum-weight triangulations, other euclidean functionals, and probabilistic recurrence relations. In: Proc. 7th Ann. ACM-SIAM Symp. on Discrete Algorithms (SODA), pp. 252–260. SIAM, Philadelphia (1996)Google Scholar
  13. 13.
    Johnson, D.S., McGeoch, L.A.: Experimental analysis of heuristics for the STSP. In: Gutin, G., Punnen, A.P. (eds.) The Traveling Salesman Problem and Its Variations, ch.9, pp. 369–443. Kluwer, Dordrecht (2002)Google Scholar
  14. 14.
    Kalpakis, K., Sherman, A.T.: Probabilistic analysis of an enhanced partitioning algorithm for the Steiner tree problem in R d. Networks 24(3), 147–159 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Karp, R.M.: Probabilistic analysis of partitioning algorithms for the traveling-salesman problem in the plane. Math. Oper. Res. 2(3), 209–224 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    León, C.A., Perron, F.: Extremal properties of sums of Bernoulli random variables. Statist. Probab. Lett. 62(4), 345–354 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Mitzenmacher, M., Upfal, E.: Probability and Computing. Cambridge University Press, Cambridge (2005)CrossRefzbMATHGoogle Scholar
  18. 18.
    Papadimitriou, C.H.: The Euclidean traveling salesman problem is NP-complete. Theoret. Comput. Sci. 4(3), 237–244 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Papadimitriou, C.H., Vazirani, U.V.: On two geometric problems related to the traveling salesman problem. J. Algorithms 5(2), 231–246 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Ravada, S., Sherman, A.T.: Experimental evaluation of a partitioning algorithm for the steiner tree problem in R 2 and R 3. Networks 24(8), 409–415 (1994)CrossRefzbMATHGoogle Scholar
  21. 21.
    Rhee, W.T.: A matching problem and subadditive euclidean functionals. Ann. Appl. Probab. 3(3), 794–801 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Röglin, H., Teng, S.-H.: Smoothed analysis of multiobjective optimization. In: Proc. 50th Ann. IEEE Symp. on Foundations of Computer Science (FOCS), pp. 681–690. IEEE, Los Alamitos (2009)Google Scholar
  23. 23.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Spielman, D.A., Teng, S.-H.: Smoothed analysis: An attempt to explain the behavior of algorithms in practice. Comm. ACM 52(10), 76–84 (2009)CrossRefGoogle Scholar
  25. 25.
    Srivastav, A., Werth, S.: Probabilistic Analysis of the Degree Bounded Minimum Spanning Tree Problem. In: Arvind, V., Prasad, S. (eds.) FSTTCS 2007. LNCS, vol. 4855, pp. 497–507. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  26. 26.
    Michael Steele, J.: Complete convergence of short paths in Karp’s algorithm for the TSP. Math. Oper. Res. 6, 374–378 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Michael Steele, J.: Subadditive Euclidean functionals and nonlinear growth in geometric probability. Ann. Probab. 9(3), 365–376 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Michael Steele, J.: Probability Theory and Combinatorial Optimization. CBMS-NSF Regional Conf. Series in Appl. Math., vol. 69. SIAM, Philadelphia (1987)Google Scholar
  29. 29.
    Supowit, K.J., Reingold, E.M.: Divide and conquer heuristics for minimum weighted euclidean matching. SIAM J. Comput. 12(1), 118–143 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Varadarajan, K.R.: A divide-and-conquer algorithm for min-cost perfect matching in the plane. In: Proc. 39th Ann. Symp. on Foundations of Computer Science (FOCS), pp. 320–331. IEEE, Los Alamitos (1998)Google Scholar
  31. 31.
    Yukich, J.E.: Probability Theory of Classical Euclidean Optimization Problems. Lecture Notes in Mathematics, vol. 1675. Springer, Heidelberg (1998)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Markus Bläser
    • 1
  • Bodo Manthey
    • 2
  • B. V. Raghavendra Rao
    • 1
  1. 1.Department of Computer ScienceSaarland UniversityGermany
  2. 2.Department of Applied MathematicsUniversity of TwenteThe Netherlands

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