Random Shortest Paths: Non-euclidean Instances for Metric Optimization Problems

  • Karl Bringmann
  • Christian Engels
  • Bodo Manthey
  • B. V. Raghavendra Rao
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8087)

Abstract

Probabilistic analysis for metric optimization problems has mostly been conducted on random Euclidean instances, but little is known about metric instances drawn from distributions other than the Euclidean.

This motivates our study of random metric instances for optimization problems obtained as follows: Every edge of a complete graph gets a weight drawn independently at random. The length of an edge is then the length of a shortest path (with respect to the weights drawn) that connects its two endpoints.

We prove structural properties of the random shortest path metrics generated in this way. Our main structural contribution is the construction of a good clustering. Then we apply these findings to analyze the approximation ratios of heuristics for matching, the traveling salesman problem (TSP), and the k-center problem, as well as the running-time of the 2-opt heuristic for the TSP. The bounds that we obtain are considerably better than the respective worst-case bounds. This suggests that random shortest path metrics are easy instances, similar to random Euclidean instances, albeit for completely different structural reasons.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Addario-Berry, L., Broutin, N., Lugosi, G.: The longest minimum-weight path in a complete graph. Combin. Probab. Comput. 19(1), 1–19 (2010)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Ausiello, G., Crescenzi, P., Gambosi, G., Kann, V., Marchetti-Spaccamela, A., Protasi, M.: Complexity and Approximation. Springer (1999)Google Scholar
  3. 3.
    Avis, D., Davis, B., Steele, J.M.: Probabilistic analysis of a greedy heuristic for Euclidean matching. Probab. Engrg. Inform. Sci. 2, 143–156 (1988)CrossRefMATHGoogle Scholar
  4. 4.
    Azar, Y.: Lower bounds for insertion methods for TSP. Combin. Probab. Comput. 3, 285–292 (1994)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Bhamidi, S., van der Hofstad, R., Hooghiemstra, G.: First passage percolation on the Erdős-Rényi random graph. Combin. Probab. Comput. 20(5), 683–707 (2011)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Blair-Stahn, N.D.: First passage percolation and competition models. arXiv:1005.0649v1 [math.PR] (2010)Google Scholar
  7. 7.
    Broadbent, S.R., Hemmersley, J.M.: Percolation processes. I. Crystals and mazes. Proceedings of the Cambridge Philosophical Society 53(3), 629–641 (1957)CrossRefMATHGoogle Scholar
  8. 8.
    Chandra, B., Karloff, H.J., Tovey, C.A.: New results on the old k-opt algorithm for the traveling salesman problem. SIAM J. Comput. 28(6), 1998–2029 (1999)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Davis, R., Prieditis, A.: The expected length of a shortest path. Inform. Process. Lett. 46(3), 135–141 (1993)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Dyer, M., Frieze, A.M., Pittel, B.: The average performance of the greedy matching algorithm. Ann. Appl. Probab. 3(2), 526–552 (1993)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Dyer, M.E., Frieze, A.M.: On patching algorithms for random asymmetric travelling salesman problems. Math. Program. 46, 361–378 (1990)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Eckhoff, M., Goodman, J., van der Hofstad, R., Nardi, F.R.: Short paths for first passage percolation on complete graphs. arXiv:1211.4569v1 [math.PR] (2012)Google Scholar
  13. 13.
    Engels, C., Manthey, B.: Average-case approximation ratio of the 2-opt algorithm for the TSP. Oper. Res. Lett. 37(2), 83–84 (2009)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Englert, M., Röglin, H., Vöcking, B.: Worst case and probabilistic analysis of the 2-opt algorithm for the TSP. In: Proc. of the 18th Ann. ACM-SIAM Symp. on Discrete Algorithms (SODA), pp. 1295–1304. SIAM (2007)Google Scholar
  15. 15.
    Frieze, A.M.: On random symmetric travelling salesman problems. Math. Oper. Res. 29(4), 878–890 (2004)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Frieze, A.M., Grimmett, G.R.: The shortest-path problem for graphs with random arc-lengths. Discrete Appl. Math. 10, 57–77 (1985)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoret. Comput. Sci. 38, 293–306 (1985)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Hassin, R., Zemel, E.: On shortest paths in graphs with random weights. Math. Oper. Res. 10(4), 557–564 (1985)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    van der Hofstad, R., Hooghiemstra, G., van Mieghem, P.: First passage percolation on the random graph. Probab. Engrg. Inform. Sci. 15(2), 225–237 (2001)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    van der Hofstad, R., Hooghiemstra, G., van Mieghem, P.: Size and weight of shortest path trees with exponential link weights. Combin. Probab. Comput. 15(6), 903–926 (2006)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Janson, S.: One, two, three times logn/n for paths in a complete graph with edge weights. Combin. Probab. Comput. 8(4), 347–361 (1999)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    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. Kluwer (2002)Google Scholar
  23. 23.
    Karp, R.M.: Probabilistic analysis of partitioning algorithms for the traveling-salesman problem in the plane. Math. Oper. Res. 2(3), 209–224 (1977)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Karp, R.M., Steele, J.M.: Probabilistic analysis of heuristics. In: Lawler, E.L., et al. (eds.) The Traveling Salesman Problem, pp. 181–205. Wiley (1985)Google Scholar
  25. 25.
    Kulkarni, V.G., Adlakha, V.G.: Maximum flow in planar networks in exponentially distributed arc capacities. Comm. Statist. Stochastic Models 1(3), 263–289 (1985)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Kulkarni, V.G.: Shortest paths in networks with exponentially distributed arc lengths. Networks 16(3), 255–274 (1986)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Kulkarni, V.G.: Minimal spanning trees in undirected networks with exponentially distributed arc weights. Networks 18(2), 111–124 (1988)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Peres, Y., Sotnikov, D., Sudakov, B., Zwick, U.: All-pairs shortest paths in o(n 2) time with high probability. In: Proc. of the 51st Ann. IEEE Symp. on Foundations of Computer Science (FOCS), pp. 663–672. IEEE (2010)Google Scholar
  29. 29.
    Reingold, E.M., Tarjan, R.E.: On a greedy heuristic for complete matching. SIAM J. Comput. 10(4), 676–681 (1981)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Rosenkrantz, D.J., Stearns, R.E., Lewis II, P.M.: An analysis of several heuristics for the traveling salesman problem. SIAM J. Comput. 6(3), 563–581 (1977)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Ross, S.M.: Introduction to Probability Models. Academic Press (2010)Google Scholar
  32. 32.
    Supowit, K.J., Plaisted, D.A., Reingold, E.M.: Heuristics for weighted perfect matching. In: Proc. of the 12th Ann. ACM Symp. on Theory of Computing (STOC), pp. 398–419. ACM (1980)Google Scholar
  33. 33.
    Vershik, A.M.: Random metric spaces and universality. Russian Math. Surveys 59(2), 259–295 (2004)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Walkup, D.W.: On the expected value of a random assignment problem. SIAM J. Comput. 8(3), 440–442 (1979)MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Yukich, J.E.: Probability Theory of Classical Euclidean Optimization Problems. Springer (1998)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Karl Bringmann
    • 1
  • Christian Engels
    • 2
  • Bodo Manthey
    • 3
  • B. V. Raghavendra Rao
    • 4
  1. 1.Max Planck Institute for InformaticsGermany
  2. 2.Saarland UniversityGermany
  3. 3.University of TwenteThe Netherlands
  4. 4.Indian Institute of Technology MadrasIndia

Personalised recommendations