Journal of Global Optimization

, Volume 53, Issue 3, pp 475–495 | Cite as

Extremal values of global tolerances in combinatorial optimization with an additive objective function

  • Vyacheslav V. Chistyakov
  • Boris I. GoldengorinEmail author
  • Panos M. Pardalos


The currently adopted notion of a tolerance in combinatorial optimization is defined referring to an arbitrarily chosen optimal solution, i.e., locally. In this paper we introduce global tolerances with respect to the set of all optimal solutions, and show that the assumption of nonembededdness of the set of feasible solutions in the provided relations between the extremal values of upper and lower global tolerances can be relaxed. The equality between globally and locally defined tolerances provides a new criterion for the multiplicity (uniqueness) of the set of optimal solutions to the problem under consideration.


Combinatorial optimization problem Additive objective function Extremal values of tolerances 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Balas E., Saltzman M.J.: An algorithm for the three-index assignment problem. Oper. Res. 39, 150–161 (1991)CrossRefGoogle Scholar
  2. 2.
    Gál T.: Postoptimal Analyses, Parametric Programming, and Related Topics. McGraw-Hill International Book Co., New York (1979)Google Scholar
  3. 3.
    Germs R., Goldengorin B., Turkensteen M.: Lower tolerance-based Branch and Bound algorithms for the ATSP. Comput. Oper. Res. 39(2), 291–298 (2012)CrossRefGoogle Scholar
  4. 4.
    Goldengorin B., Jager G., Molitor P.: Tolerances applied in combinatorial optimization. J. Comput. Sci. 2(9), 716–734 (2006)CrossRefGoogle Scholar
  5. 5.
    Goldengorin, B., Sierksma, G.: Combinatorial optimization tolerances calculated in linear time. SOM Research Report 03a30, University of Groningen, The Netherlands, pp 1–6 (2003)Google Scholar
  6. 6.
    Goldengorin B., Sierksma G., Turkensteen M.: Tolerance based algorithms for the ATSP. Lect. Notes Comput. Sci. 3353, 222–234 (2004)CrossRefGoogle Scholar
  7. 7.
    Greenberg, H.: An annotated bibliography for post-solution analysis in mixed integer programming and combinatorial optimization. In: Advances in Computational and Stochastic Optimization, Logic Programming, and Heuristic Search, pp. 97–147. Kluwer, Boston (1998)Google Scholar
  8. 8.
    Gusfield D.: A note on arc tolerances in sparse shortest-path and network flow problems. Networks 13(2), 191–196 (1983)CrossRefGoogle Scholar
  9. 9.
    Gutin G., Goldengorin B., Huang J.: Worst case analysis of GREEDY, Max-Regret and other heuristics for multidimensional assignment and traveling salesman problems. J. Heurist. 14(2), 169–181 (2008)CrossRefGoogle Scholar
  10. 10.
    Helsgaun K.: An effective implementation of the Lin-Kernighan traveling salesman heuristic. Eur. J. Oper. Res. 126(1), 106–130 (2000)CrossRefGoogle Scholar
  11. 11.
    Jäger G.: The Theory of Tolerances with Applications to the Traveling Salesman Problem. Habilitationsschrift, Kiel (2010)Google Scholar
  12. 12.
    Libura M.: Sensitivity analysis for minimum Hamiltonian path and traveling salesman problems. Discrete Appl. Math. 30(2–3), 197–211 (1991)CrossRefGoogle Scholar
  13. 13.
    Libura M.: A note on robustness tolerances for combinatorial optimization problems. Inf. Process. Lett. 110(16), 725–729 (2010)CrossRefGoogle Scholar
  14. 14.
    Little J.D.C., Murty K.G., Sweeny W.W., Karel C.: An algorithm for the traveling salesman problem. Oper. Res. 11, 972–989 (1963)CrossRefGoogle Scholar
  15. 15.
    Murty K.G.: An algorithm for ranking all the assignments in order of increasing cost. Oper. Res. 16, 682–687 (1968)CrossRefGoogle Scholar
  16. 16.
    Pardalos P.M., Jha S.: Complexity of uniqueness and local search in quadratic 0-1 programming. Oper. Res. Let. 11, 119–123 (1992)CrossRefGoogle Scholar
  17. 17.
    Reinfeld N.V., Vogel W.R.: Mathematical Programming. Prentice-Hall, Englewood Cliffs (1958)Google Scholar
  18. 18.
    Reinelt G.: The Linear Ordering Problem: Algorithms and Applications. Heldermann Verlag, Berlin (1985)Google Scholar
  19. 19.
    Shier D.R., Witzgall C.: Arc tolerances in shortest path and network flow problems. Networks 10(4), 277–291 (1980)CrossRefGoogle Scholar
  20. 20.
    Tarjan R.E.: Sensitivity analysis of minimum spanning trees and shortest path trees. Inf. Precess. Lett. 14(1), 30–33 (1982)CrossRefGoogle Scholar
  21. 21.
    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)CrossRefGoogle Scholar
  22. 22.
    Van Hoesel S., Wagelmans A.P.M.: On the complexity of postoptimality analysis of 0/1 programs. Discrete Appl. Math. 91(1–3), 251–263 (1999)CrossRefGoogle Scholar
  23. 23.
    van der Poort E.S., Libura M., Sierksma G., van der Veen J.A.A.: Solving the k-best traveling salesman problem. Comput. Oper. Res. 26(4), 409–425 (1999)CrossRefGoogle Scholar
  24. 24.
    Zhang W., Korf R.E.: A study of complexity transitions on the asymmetric traveling salesman problem. Artif. Intell. 81(1–2), 223–239 (1996)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC. 2012

Authors and Affiliations

  • Vyacheslav V. Chistyakov
    • 1
    • 2
  • Boris I. Goldengorin
    • 1
    • 2
    Email author
  • Panos M. Pardalos
    • 2
    • 3
  1. 1.Department of Applied Mathematics and Computer ScienceNational Research University Higher School of EconomicsNizhny NovgorodRussian Federation
  2. 2.Laboratory of Algorithms and Technologies for Networks AnalysisNational Research University Higher School of EconomicsNizhny NovgorodRussian Federation
  3. 3.Department of Industrial and Systems Engineering, Center for Applied OptimizationUniversity of FloridaGainesvilleUSA

Personalised recommendations