Advertisement

Journal of Global Optimization

, Volume 68, Issue 3, pp 601–622 | Cite as

The reduction of computation times of upper and lower tolerances for selected combinatorial optimization problems

  • Marcel Turkensteen
  • Dmitry Malyshev
  • Boris Goldengorin
  • Panos M. Pardalos
Article
  • 184 Downloads

Abstract

The tolerance of an element of a combinatorial optimization problem with respect to its optimal solution is the maximum change of the cost of the element while preserving the optimality of the given optimal solution and keeping all other input data unchanged. Tolerances play an important role in the design of exact and approximation algorithms, but the computation of tolerances requires additional computational time. In this paper, we concentrate on combinatorial optimization problems for which the computation of all tolerances and an optimal solution have almost the same computational complexity as of finding an optimal solution only. We summarize efficient computational methods for computing tolerances for these problems and determine their time complexity experimentally.

Keywords

Discrete optimization Tolerances Complexity Efficient algorithm 

Notes

Acknowledgements

The research of D.S. Malyshev and P.M. Pardalos is partially supported by LATNA laboratory, National Research University Higher School of Economics. B. Goldengorin’s research is supported by C. Paul Stocker Visiting Professorship provided by the Department of Industrial and Systems Engineering, The Russ College of Engineering, Ohio University, Athens, OH, USA.

References

  1. 1.
    Achterberg, T., Koch, T., Martin, A.: Branching rules revisited. Oper. Res. Lett. 33(1), 42–54 (2004)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Almoustafa, S., Hanafi, S., Mladenovic, N.: New exact method for large asymmetric distance-constrained vehicle routing problem. Eur. J. Oper. Res. 226, 386–394 (2013)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Balas, E., Toth, P.: Branch and bound methods. In: Lawler, E.L., Lenstra, J.K., Rinnooy Kan, A.H.G., Shmoys, D.B. (eds.) Chapter 10 of the Traveling Salesman Problem, pp. 361–401. Wiley, Chichester (1985)Google Scholar
  4. 4.
    Batsyn, M., Goldengorin, B.I., Kocheturov, A., Pardalos, P.M.: Tolerance-based versus cost-based branching for the asymmetric capacitated vehicle routing problem. In: Goldengorin, B.I. et al. (eds.) Models, Algorithms, and Technologies for Network Analysis. Proceedings in Mathematics and Statistics, vol. 59, Springer, Berlin (2013)Google Scholar
  5. 5.
    Bekker, H., Braad, E.P., Goldengorin, B.: Using bipartite and multidimensional matching to select the roots of a system of polynomial equations. Lect. Notes Comput. Sci. 3483, 397–406 (2005)CrossRefMATHGoogle Scholar
  6. 6.
    Bekker, H., Braad, E.P., Goldengorin, B.: Selecting the roots of a small system of polynomial equations by tolerance-based matching. Lect. Notes Comput. Sci. 3503, 610–613 (2005)CrossRefMATHGoogle Scholar
  7. 7.
    Booth, H., Westbrook, J.: A linear algorithm for analysis of minimum spanning and shortest-path trees of planar graphs. Algorithmica 11, 341–352 (1994)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Chen, C., Kuo, M., Sheu, J.: An optimal time algorithm for finding a maximum weight independent set in a tree. BIT Numer. Math. 28, 353–356 (1988)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Chin, F., Houck, D.: Algorithms for updating minimal spanning trees. J. Comput. Syst. Sci. 16, 333–344 (1978)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Cormen, T.H., Leiserson, C.E., Rivest, R.L., Stein, C.: Section 24.3: Dijkstra’s Algorithm in Introduction to Algorithms, 2nd edn, pp. 595–601. MIT Press and McGraw Hill, NewYork (2001)Google Scholar
  11. 11.
    Dell’Amico, M., Toth, P.: Algorithms and codes for dense assignment problems: the state of the art. Discrete Appl. Math. 140, 1–3 (2004)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Dijkstra, E.W.: A note on two problems in connexion with graphs. Numer. Math. 1(1), 269–271 (1959)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    DIMACS, Benchmark instance generates for the ATSP. http://dimacs.rutgers.edu/Challenges/TSP/atsp.html (2006)
  14. 14.
    Dixon, B., Rauch, M., Tarjan, R.E.: Verification and sensitivity analysis of minimum spanning trees in linear time. SIAM J. Comput. 21(6), 1184–1192 (1992)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Fischetti, M., Toth, P., Vigo, D.: A branch-and-bound algorithm for the capacitated vehicle routing problem on directed graphs. Oper. Res. 42(5), 846–859 (1994)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Fisher, M.L.: The Lagrangian relaxation method for solving integer programming problems. Manag. Sci. 27(1), 1–18 (1981)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Germs, R., Goldengorin, B., Turkensteen, M.: Lower tolerance-based branch and bound algorithms for the ATSP. Comput. Oper. Res. 39(2), 291–298 (2012)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Ghosh, D., Goldengorin, B., Gutin, G., Jäger, G.: Tolerance-based algorithms for the traveling salesman problem. In: Neogy, S.K., Bapat, R.B., Das, A.K., Parthasarathy, T. (eds.) Mathematical Programming and Game Theory for Decision Making. Statistical Science Interdisciplinary Research, 47–59, World Scientific Publication, Hackensack, NJ (2008)Google Scholar
  19. 19.
    Ghosh, D., Goldengorin, B., Gutin, G., Jäger, G.: Tolerance-based algorithms for the traveling salesman problem. In: Neogy, S.K., Bapat, R.B., Das, A.K., Parthasarathy, T. (eds.) Chapter of Mathematical Programming and Game Theory for Decision Making, pp. 47–59. World Scientific, New Jersey (2008)CrossRefGoogle Scholar
  20. 20.
    Goldengorin, B., Jäger, G., Molitor, P.: Some Basics on Tolerances. In: Cheng, S.-W., Poon, C.K. (eds.) Proceedings of the 2nd International Conference on Algorithmic Aspects in Information and Management (AAIM). Lecture Notes in Computer Science 4041, 194–206 (2006)Google Scholar
  21. 21.
    Goldengorin, B., Jäger, G. Molitor, P.: Tolerance-based Contract-or-Patch Heuristic for the Asymmetric TSP. In: Erlebach, T. (ed.) Proceedinggs 3rd Workshop on Combinatorial and Algorithmic Aspects of Networking (CAAN). Lecture Notes in Computer Science 4235, 86–97 (2006)Google Scholar
  22. 22.
    Goldengorin, B., Sierksma, G., Turkensteen, M.: Tolerance-based algorithms for the ATSP. In: Hromkovic, J., Nagl, M., Westfechtel, B. (eds.) Proceedings of the 30th International Workshop on Graph-Theoretic Concepts in Computer Science (WG). Lecture Notes in Computer Science 3353, 222–234 (2004)Google Scholar
  23. 23.
    Goldengorin, B., Sierksma, G.: Combinatorial Optimization Tolerances Calculated in Linear Time. Research Report 03A30, Graduate School/Research Institute Systems, Organizations and Management, University of Groningen, Groningen, The Netherlands (2003)Google Scholar
  24. 24.
    Goldengorin, B., Malyshev, D.S., Pardalos, P.M.: Efficient computation of tolerances in weighted independent set problems for trees. Dokl. Math. 87(3), 368–371 (2013)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Goldengorin, B., Malyshev, D.S., Pardalos, P.M., Zamaraev, V.A.: A tolerance-based heuristic for the weighted independent set problem. J. Comb. Optim. 29, 433–450 (2015)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Graham, R.L., Hell, P.: On the history of the minimum spanning tree problem. Ann. Hist. Comput. 7(1), 43–57 (1985)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Held, M., Karp, R.: The traveling-salesman problem and minimum spanning trees. Oper. Res. 18, 1138–1162 (1970)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Helsgaun, K.: An effective implementation of the Lin–Kernighan traveling salesman heuristic. Eur. J. Oper. Res. 126(1), 106–130 (2000)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Hillier, F.S., Lieberman, G.J.: Introduction to Operations Research, 6th edn. McGraw Hill, Singapore (1995)MATHGoogle Scholar
  30. 30.
    Johnson, D.S., Gutin, G., McGeoch, L.A., Yeo, A., Zhang, W., Zverovich, A.: Experimental analysis of heuristics for the ATSP. In: Gutin, G., Punnen, A.P. (eds.) The Traveling Salesman Problem and its Variations, pp. 445–487. Kluwer Academic Publishers, Berlin (2002)Google Scholar
  31. 31.
    Jonker, R., Volgenant, A.: Assignment Solver Code. http://www.assignmentproblems.com/LAPJV.htm (1986)
  32. 32.
    Jonker, R., Volgenant, A.: Improving the Hungarian assignment algorithm. Oper. Res. Lett. 5, 171–175 (1986)MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Jonker, R., Volgenant, A.: A shortest augmenting path algorithm for dense and sparse linear assignment problems. Computing 38, 325–340 (1987)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Kindervater, G., Volgenant, A., de Leve, G., van Gijlswijk, V.: On dual solutions of the linear assignment problem. Eur. J. Oper. Res. 19(1), 76–81 (1985)MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Kruskal, J.B.: On the shortest spanning tree of a graph and the traveling salesman problem. Proc. Am. Soc. 7, 48–50 (1956)CrossRefMATHGoogle Scholar
  36. 36.
    Kuhn, H.W.: The Hungarian method for the assignment problem. Naval Res. Logist. Q. 2, 83–97 (1955)MathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    Lin, C.J., Wen, U.P.: Sensitivity analysis of the optimal assignment. Eur. J. Oper. Res. 149(1), 35–46 (2003)MathSciNetCrossRefMATHGoogle Scholar
  38. 38.
    Malyshev, D.S., Pardalos, P.M.: Efficient computation of tolerances in the weighted independent set problem for some classes of graphs. Dokl. Math. 89(2), 253–256 (2014)MathSciNetCrossRefMATHGoogle Scholar
  39. 39.
    Pettie, S.: Sensitivity analysis of minimum spanning trees in sub-inverse-ackermann time. In: Deng, X., Du, D., (eds.): ISAAC 2005, LNCS 3827 964–973 (2005)Google Scholar
  40. 40.
    Pettie, S., Ramachandran, V.: An optimal minimum spanning tree algorithm. J. ACM 49(1), 16–34 (2002)MathSciNetCrossRefMATHGoogle Scholar
  41. 41.
    Prim, R.C.: Shortest connection networks and some generalizations. Bell Syst. Technol. J. 36, 1389–1401 (1957)CrossRefGoogle Scholar
  42. 42.
    Ramaswamy, R., Orlin, J.B., Chakravarti, N.: Sensitivity analysis for shortest path problems and maximum capacity path problems in undirected graphs. Math. Program. Ser. A 102, 355–369 (2005)MathSciNetCrossRefMATHGoogle Scholar
  43. 43.
    Reinelt, G.: TSPLIB—a traveling salesman problem library. ORSA J. Comput. 3, 376–384 (1991)CrossRefMATHGoogle Scholar
  44. 44.
    Reinfeld, N.V., Vogel, W.R.: Mathematical Programming. Prentice-Hall, Englewood Cliffs, NJ (1958)Google Scholar
  45. 45.
    Richter, D., Goldengorin, B., Jager, G., Molitor, P.: Improving the efficiency of Helsgaun’s Lin–Kernighan heuristic for the symmetric TSP. Lect. Notes Comput. Sci. 4852, 99–111 (2007)CrossRefMATHGoogle Scholar
  46. 46.
    Salles da Cunha, A., Lucena, A.: Lower and upper bounds for the degree-constrained minimum spanning tree problem. Networks 50(1), 55–66 (2007)MathSciNetCrossRefMATHGoogle Scholar
  47. 47.
    Shier, D.R., Witzgall, C.: Arc tolerances in minimum-path and network flow problems. Networks 10, 277–291 (1980)MathSciNetCrossRefMATHGoogle Scholar
  48. 48.
    Tarjan, R.E.: Sensitivity analysis of minimum spanning trees and shortest path trees. Inf. Process. Lett. 14(1), 30–33 (1982)MathSciNetCrossRefGoogle Scholar
  49. 49.
    Toth, P.: Optimization engineering techniques for the exact solution of NP-hard combinatorial optimization problems. Eur. J. Oper. Res. 125, 222–238 (2000)MathSciNetCrossRefMATHGoogle Scholar
  50. 50.
  51. 51.
    Turkensteen, M., Ghosh, D., Goldengorin, B., Sierksma, G.: Tolerance-based branch and bound algorithms for the ATSP. Eur. J. Oper. Res. 189(3), 775–788 (2008)MathSciNetCrossRefMATHGoogle Scholar
  52. 52.
    Volgenant, A.: A Lagrangean approach to the degree-constrained minimum spanning tree problem. Eur. J. Oper. Res. 39, 325–331 (1989)MathSciNetCrossRefMATHGoogle Scholar
  53. 53.
    Volgenant, A.: An addendum on sensitivity analysis of the optimal assignment. Eur. J. Oper. Res. 169, 338–339 (2006)CrossRefMATHGoogle Scholar
  54. 54.
    Volgenant, T., Jonker, R.: The symmetric traveling salesman problem and edge exchanges in minimal 1-trees. Eur. J. Oper. Res. 12, 394–403 (1983)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  • Marcel Turkensteen
    • 1
  • Dmitry Malyshev
    • 2
  • Boris Goldengorin
    • 3
    • 4
  • Panos M. Pardalos
    • 4
  1. 1.Aarhus UniversityAarhus VDenmark
  2. 2.National Research University Higher School of EconomicsNizhny NovgorodRussia
  3. 3.Ohio UniversityAthensUSA
  4. 4.University of FloridaGainesvilleUSA

Personalised recommendations