Some Basics on Tolerances

  • Boris Goldengorin
  • Gerold Jäger
  • Paul Molitor
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4041)


In this paper we deal with sensitivity analysis of combinatorial optimization problems and its fundamental term, the tolerance. For three classes of objective functions (\(\Sigma, \Pi, {\mbox{MAX}}\)) we give some basic properties on upper and lower tolerances. We show that the upper tolerance of an element is well defined, how to compute the upper tolerance of an element, and give equivalent formulations when the upper tolerance is +∞ or > 0. Analogous results are given for the lower tolerance and some results on the relationship between lower and upper tolerances are given.


Sensitivity analysis upper tolerance lower tolerance 


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)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Bang-Jensen, J., Gutin, G.: Digraphs: Theory, Algorithms and Applications. Springer, London (2002)zbMATHGoogle Scholar
  3. 3.
    Chin, F., Hock, D.: Algorithms for Updating Minimal Spanning Trees. J. Comput. System Sci. 16, 333–344 (1978)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Gal, T.: Sensitivity Analysis, Parametric Programming, and Related Topics: Degeneracy, Multicriteria Decision Making, Redundancy. W. de Gruyter, Berlin (1995)Google Scholar
  5. 5.
    Gal, T., Greenberg, H.J. (eds.): Advances in Sensitivity Analysis and Parametric Programming. Internat. Ser. Oper. Res. Management Sci., vol. 6. Kluwer Academic Publishers, Boston (1997)zbMATHGoogle Scholar
  6. 6.
    Goldengorin, B., Jäger, G.: How To Make a Greedy Heuristic for the Asymmetric Traveling Salesman Competitive. SOM Research Report 05A11, University of Groningen, The Netherlands (2005),
  7. 7.
    Goldengorin, B., Jäger, G., Molitor, P.: Some Basics on Tolerances. SOM Research Report 05A13, University of Groningen, The Netherlands (2005),
  8. 8.
    Goldengorin, B., Sierksma, G.: Combinatorial optimization tolerances calculated in linear time. SOM Research Report 03A30, University of Groningen, The Netherlands (2003),
  9. 9.
    Goldengorin, B., Sierksma, G., Turkensteen, M.: Tolerance Based Algorithms for the ATSP. In: Hromkovič, J., Nagl, M., Westfechtel, B. (eds.) WG 2004. LNCS, vol. 3353, pp. 222–234. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  10. 10.
    Gordeev, E.N., Leontev, V.K., Sigal, I.K.: Computational algorithms for finding the radius of stability in selection problems. USSR Comput. Math. Math. Phys. 23, 973–979 (1983)zbMATHCrossRefGoogle Scholar
  11. 11.
    Greenberg, H.J.: An annotated bibliography for post-solution analysis in mixed integer and combinatorial optimization. In: Woodruff, D.L. (ed.) Advances in Computational and Stochastic Optimization, Logic Programming, and Heuristic Search, pp. 97–148. Kluwer Academic Publishers, Dordrecht (1998)Google Scholar
  12. 12.
    Gusfield, D.: A note on arc tolerances in sparse minimum-path and network flow problems. Networks 13, 191–196 (1983)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Helsgaun, K.: An effective implementation of the Lin-Kernighan traveling salesman heuristic. European J. Oper. Res. 126, 106–130 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Kravchenko, S.A., Sotskov, Y.N., Werner, F.: Optimal schedules with infinitely large stability radius. Optimization 33, 271–280 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Libura, M.: Sensitivity analysis for minimum hamiltonian path and traveling salesman problems. Discrete Appl. Math. 30, 197–211 (1991)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Murty, K.G.: An algorithm for ranking all the assignments in order of increasing cost. Oper. Res. 16, 682–687 (1968)zbMATHCrossRefGoogle Scholar
  17. 17.
    Ramaswamy, R., Chakravarti, N.: Complexity of determining exact tolerances for min-sum and min-max combinatorial optimization problems. Working Paper WPS-247/95, Indian Institute of Management, Calcutta, India, p. 34 (1995)Google Scholar
  18. 18.
    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)zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Reinfeld, N.V., Vogel, W.R.: Mathematical Programming. Prentice-Hall, Englewood Cliffs (1958)Google Scholar
  20. 20.
    Shier, D.R., Witzgall, C.: Arc tolerances in minimum-path and network flow problems. Networks 10, 277–291 (1980)CrossRefMathSciNetGoogle Scholar
  21. 21.
    Sotskov, Y.N.: The stability of the approximate boolean minimization of a linear form. USSR Comput. Math. Math. Phys. 33, 699–707 (1993)MathSciNetGoogle Scholar
  22. 22.
    Sotskov, Y.N., Leontev, V.K., Gordeev, E.N.: Some concepts of stability analysis in combinatorial optimization. Discrete Appl. Math. 58, 169–190 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Tarjan, R.E.: Sensitivity Analysis of Minimum Spanning Trees and Shortest Path Trees. Inform. Process. Lett. 14(1), 30–33 (1982)CrossRefMathSciNetGoogle Scholar
  24. 24.
    Turkensteen, M., Ghosh, D., Goldengorin, B., Sierksma, G.: Tolerance-Based Branch and Bound Algorithms. In: Maroto, C., et al. (eds.) Proceedings of a EURO conference for young OR researches and practitioners, ORP3 2005, Valencia, Spain, September 6–10, pp. 171–182 (2005)Google Scholar
  25. 25.
    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, 409–425 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Van Hoesel, S., Wagelmans, A.: On the complexity of postoptimality analysis of 0/1 programs. Discrete Appl. Math. 91, 251–263 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Volgenant, A.: An addendum on sensitivity analysis of the optimal assignment. European J. Oper. Res. 169, 338–339 (2006)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Boris Goldengorin
    • 2
    • 3
  • Gerold Jäger
    • 1
  • Paul Molitor
    • 1
  1. 1.Computer Science InstituteUniversity of Halle-WittenbergHalle (Saale)Germany
  2. 2.Faculty of Economic SciencesUniversity of GroningenGroningenThe Netherlands
  3. 3.Department of Applied MathematicsKhmelnitsky National UniversityUkraine

Personalised recommendations