Journal of Heuristics

, Volume 16, Issue 5, pp 633–651 | Cite as

Heuristics for the central tree problem

  • Jørgen Bang-Jensen
  • Yury Nikulin


This paper addresses the central spanning tree problem (CTP). The problem consists in finding a spanning tree that minimizes the so-called robust deviation, i.e. deviation from a maximally distant tree. The distance between two trees is measured by means of the symmetric difference of their edge sets. The central tree problem is known to be NP-hard. We attack the problem with a hybrid heuristic consisting of: (1) a greedy construction heuristic to get a good initial solution and (2) fast local search improvement. We illustrate computationally efficiency of the proposed approach.


Minmax regret Robustness Central tree Spanning tree Local search 


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  1. Amoia, A., Cottafava, G.: Invariance properties of central trees. IEEE Trans. Circuit Theory CT 18, 465–467 (1971) CrossRefMathSciNetGoogle Scholar
  2. Aron, I., van Hentenryck, P.: A constraint satisfaction approach to the robust spanning tree with interval data. In: Proceedings of the 18th Conference on UAI, Edmonton, Canada, August 1–4, 2002 Google Scholar
  3. Aron, I., van Hentenryck, P.: On the complexity of the robust spanning tree problem with interval data. Oper. Res. Lett. 32, 36–40 (2004) zbMATHCrossRefMathSciNetGoogle Scholar
  4. Averbakh, I., Lebedev, V.: Interval data regret network optimization problems. Discrete Appl. Math. 138, 289–301 (2004) zbMATHCrossRefMathSciNetGoogle Scholar
  5. Bang-Jensen, J., Gutin, G.: Digraphs: Theory, Algorithms and Applications. Springer, London-Berlin-Heidelberg (2001) zbMATHGoogle Scholar
  6. Bezrukov, S., Kaderali, F., Poguntke, W.: On central spanning trees of a graph. Lect. Notes Comput. Sci. 1120, 53–58 (1996) MathSciNetGoogle Scholar
  7. Deo, N.: A central tree. IEEE Trans. Circuit Theory CT 13, 439–440 (1966) CrossRefMathSciNetGoogle Scholar
  8. Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. Freeman, New York (1979) zbMATHGoogle Scholar
  9. Kaderali, F.: A counterexample to the algorithm of Amoia and Cottafava for finding central trees. Technical Report FB 19, TH Darmstadt, June 1973, Preprint Google Scholar
  10. Kasperski, A., Zielinski, P.: An approximation algorithm for interval data minmax regret combinatorial optimization problems. Inf. Process. Lett. 97, 177–180 (2006) zbMATHCrossRefMathSciNetGoogle Scholar
  11. Kishi, G., Kajitani, Y.: Maximally distant trees and principal partition of a linear graph. IEEE Trans. Circuit Theory CT 16, 323–330 (1969) CrossRefMathSciNetGoogle Scholar
  12. Kouvelis, P., Yu, G.: Robust Discrete Optimization and Its Applications. Kluwer Academic, Norwell (1997) zbMATHGoogle Scholar
  13. Kozina, G., Perepelitsa, V.: Interval spanning tree problem: solvability and computational complexity. Interval Comput. 1, 42–50 (1994) MathSciNetGoogle Scholar
  14. Lee, Y.Y.: Use of compound matrices in topological analysis. Master‘s Thesis, Kansas State University, Manhatten, KS (1968) Google Scholar
  15. Montemanni, R.: A Benders decomposition approach for the robust spanning tree problem with interval data. Eur. J. Oper. Res. 174, 1479–1490 (2006) zbMATHCrossRefMathSciNetGoogle Scholar
  16. Montemanni, R., Gambardella, L.: A branch and bound algorithm for the robust spanning tree problem with interval data. Eur. J. Oper. Res. 161, 771–779 (2005) zbMATHCrossRefMathSciNetGoogle Scholar
  17. Nagamochi, H., Ibaraki, T.: Computing edge-connectivity in multigraphs and capacitated graphs. SIAM J. Discrete Math. 5, 54–66 (1992) zbMATHCrossRefMathSciNetGoogle Scholar
  18. Nikulin, Y.: Robustness in combinatorial optimization and scheduling theory: an extended annotated bibliography. Manuskripte aus den Instituten für Betriebswirtschaftslehre No. 606, Christian-Albrechts-Universität zu Kiel, Germany (2006) Google Scholar
  19. Nikulin, Y.: Simulated annealing algorithm for the robust spanning tree problem. J. Heuristics 14, 391–402 (2008) zbMATHCrossRefGoogle Scholar
  20. Shinoda, S., Kawamoto, T.: Central trees and critical sets. In: Kirk, D.E. (ed.) Proceedings of the 14th Asilomar Conference on Circuit, Systems and Computers, vol. 108, Pacific Grove, California, pp. 183–187 (1980) Google Scholar
  21. Shinoda, S., Kawamoto, T.: On central trees of a graph. In: Lecture Notes in Computer Science, vol. 108, pp. 137–151. Springer, Berlin (1981) Google Scholar
  22. Shinoda, S., Saishu, K.: Conditions for an incidence set to be a central tree. Technical Report CAS80-6, Institute of Electrical and Communication Engineering, Japan (1980) Google Scholar
  23. Stoer, M., Wagner, F.: A simple mincut algorithm. In: Proc. of ESA94. Lecture Notes in Computer Science, vol. 855, pp. 141–147 (1994) Google Scholar
  24. Tutte, W.T.: On the problem of decomposing a graph into n connected factors. J. Lond. Math. Soc. 142, 221–230 (1961) CrossRefMathSciNetGoogle Scholar
  25. Yaman, H., Karasan, O., Pinar, M.: The robust spanning tree problem with interval data. Oper. Res. Lett. 29, 31–40 (2001) zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.Department of Mathematics and Computer ScienceUniversity of Southern DenmarkOdenseDenmark
  2. 2.Department of MathematicsUniversity of TurkuTurkuFinland

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