Skip to main content
SpringerLink
Log in

A tabu search algorithm for the minmax regret minimum spanning tree problem with interval data

  • Published:
Journal of Heuristics Aims and scope Submit manuscript

Abstract

This paper deals with the strongly NP-hard minmax regret version of the minimum spanning tree problem with interval costs. The best known exact algorithms solve the problem in reasonable time for rather small graphs. In this paper an algorithm based on the idea of tabu search is constructed. Some properties of the local minima are shown. Exhaustive computational tests for various classes of graphs are performed. The obtained results suggest that the proposed tabu search algorithm quickly outputs optimal solutions for the smaller instances, previously discussed in the existing literature. Furthermore, some arguments that this algorithm performs well also for larger instances are provided.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9

Similar content being viewed by others

References

  • Ahuja, R.K., Magnanti, T.L., Orlin, J.B.: Network Flows, Theory, Algorithms and Applications. Prentice Hall, Englewood Cliffs (1993)

    MATH  Google Scholar 

  • Ahuja, R.K., Ergun, O., Orlin, J., Punnen, A.P.: A survey of very large-scale neighborhood search techniques. Discrete Appl. Math. 123, 75–102 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  • Amoia, A., Cottafava, G.: Invariance properties of central trees. IEEE Trans. Circuit Theory CT-18, 465–467 (1971)

    Article  MathSciNet  Google Scholar 

  • Aron, I., van Hentenryck, P.: A constraint satisfaction approach to the robust spanning tree with interval data. In: Proceedings of the 18th Conference on Uncertainty in Artificial Intelligence, Edmonton, Canada (2002)

    Google Scholar 

  • Aron, I., van Hentenryck, P.: On the complexity of the robust spanning tree problem with interval data. Oper. Res. Lett. 32, 36–40 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  • Averbakh, I., Lebedev, V.: Interval data minmax regret network optimization problems. Discrete Appl. Math. 138, 289–301 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  • Bang-Jensen, J., Nikulin, J.: Heuristics for the central tree problem. J. Heuristics 16, 633–651 (2010)

    Article  MATH  Google Scholar 

  • Bezrukov, S., Kaderali, F., Poguntke, W.: On central spanning trees of a graph. In: Lectures Notes in Computer Science, vol. 1120, pp. 53–57. Springer, Berlin (1995)

    Google Scholar 

  • Chazelle, B.: A minimum spanning tree algorithm with inverse-Ackermann type complexity. J. ACM 47, 1028–1047 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  • Cormen, T.H., Leiserson, C.E., Rivest, R.L., Stein, C.: Introduction to Algorithms, 2nd edn. The MIT Press and McGraw-Hill, Cambridge, New York (2001)

    MATH  Google Scholar 

  • Deo, D.: A central tree. IEEE Trans. Circuit Theory CT-13, 439–440 (1966)

    Article  MathSciNet  Google Scholar 

  • Glover, F.: Tabu Search Part I. ORSA J. Comput. 1(3), 190–206 (1989)

    Article  MATH  Google Scholar 

  • Glover, F.: Tabu Search Part II. ORSA J. Comput. 2(1), 4–32 (1990)

    Article  MATH  Google Scholar 

  • Glover, F., Laguna, M.: Tabu Search. Kluwer, Norwell (1997)

    Book  MATH  Google Scholar 

  • Kasperski, A., Zieliński, P.: An approximation algorithm for interval data minmax regret combinatorial optimization problems. Inf. Process. Lett. 97(5), 177–180 (2006)

    Article  MATH  Google Scholar 

  • Kasperski, A., Zieliński, P.: On combinatorial optimization problems on matroids with ill-known weights. Eur. J. Oper. Res. 177, 851–864 (2007)

    Article  MATH  Google Scholar 

  • Kasperski, A., Kobylański, P., Kulej, M., Zieliński, P.: Minimizing maximal regret in discrete optimization problems with interval data. In: Hryniewicz, O., Kacprzyk, J., Kuchta, D. (eds.) Issues in Soft Computing Decisions and Operations Research, pp. 193–208. Akademicka Oficyna Wydawnicza EXIT, Warsaw (2005)

    Google Scholar 

  • Kouvelis, P., Yu, G.: Robust Discrete Optimization and Its Applications. Kluwer Academic, Boston (1997)

    MATH  Google Scholar 

  • Kruskal, J.B.: On the shortest spanning subtree of a graph and the traveling salesman problem. Proc. Am. Math. Soc. 7, 48–50 (1956)

    Article  MathSciNet  MATH  Google Scholar 

  • Lawler, E.L.: Combinatorial Optimization: Networks and Matroids. Holt, Rinehart and Winston, New York (1976)

    MATH  Google Scholar 

  • Michiels, W., Aarts, E., Korst, J.: Theoretical Aspects of Local Search. Springer, Berlin (2007)

    MATH  Google Scholar 

  • Montemanni, R., Gambardella, L.M., Donati, A.V.: A branch and bound algorithm for the robust shortest path problem with interval data. Oper. Res. Lett. 32(3), 225–232 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  • Montemanni, R., Gambardella, L.M.: A branch and bound algorithm for the robust spanning tree problem with interval data. Eur. J. Oper. Res. 161, 771–779 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  • Montemanni, R.: A Benders decomposition approach for the robust spanning tree problem with interval data. Eur. J. Oper. Res. 174(3), 1479–1490 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  • Montemanni, R., Barta, J., Gambardella, L.: The robust traveling salesman problem with interval data. Transp. Sci. 41, 366–381 (2007)

    Article  Google Scholar 

  • Nikulin, Y.: Simulated annealing algorithm for the robust spanning tree problem. J. Heuristics 14, 391–402 (2008)

    Article  MATH  Google Scholar 

  • Oxley, J.G.: Matroid Theory. Oxford University Press, New York (1992)

    MATH  Google Scholar 

  • Papadimitriou, C., Steiglitz, K.: Combinatorial Optimization, Algorithms and Complexity. Dover, Mineola (1998)

    MATH  Google Scholar 

  • Prim, R.C.: Shortest connection networks and some generalizations. Bell Syst. Tech. J. 36, 1389–1401 (1957)

    Google Scholar 

  • Ribeiro, C., Uchoa, E., Werneck, R.: A hybrid GRASP with perturbations for the Steiner problem in graphs. INFORMS J. Comput. 14, 228–246 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  • Yaman, H., Karasan, O.E., Pinar, M.C.: The robust spanning tree with interval data. Oper. Res. Lett. 29, 31–40 (2001)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Industrial Engineering and Management, Wrocław University of Technology, Wybrzeże Wyspiańskiego 27, 50-370, Wrocław, Poland

    Adam Kasperski

  2. Institute of Computer Engineering, Control and Robotics, Wrocław University of Technology, Wybrzeże Wyspiańskiego 27, 50-370, Wrocław, Poland

    Mariusz Makuchowski

  3. Institute of Mathematics and Computer Science, Wrocław University of Technology, Wybrzeże Wyspiańskiego 27, 50-370, Wrocław, Poland

    Paweł Zieliński

Authors
  1. Adam Kasperski
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mariusz Makuchowski
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Paweł Zieliński
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Adam Kasperski.

Additional information

This work was partially supported by Polish Committee for Scientific Research, grant N N206 492938.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kasperski, A., Makuchowski, M. & Zieliński, P. A tabu search algorithm for the minmax regret minimum spanning tree problem with interval data. J Heuristics 18, 593–625 (2012). https://doi.org/10.1007/s10732-012-9200-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10732-012-9200-z

Keywords

Search

Navigation