Advertisement

A Primal Branch-and-Cut Algorithm for the Degree-Constrained Minimum Spanning Tree Problem

  • Markus Behle
  • Michael Jünger
  • Frauke Liers
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4525)

Abstract

The degree-constrained minimum spanning tree (DCMST) is relevant in the design of networks. It consists of finding a spanning tree whose nodes do not exceed a given maximum degree and whose total edge length is minimum. We design a primal branch-and-cut algorithm that solves instances of the problem to optimality. Primal methods have not been used extensively in the past, and their performance often could not compete with their standard ‘dual’ counterparts. We show that primal separation procedures yield good bounds for the DCMST problem. On several instances, the primal branch-and-cut program turns out to be competitive with other methods known in the literature. This shows the potential of the primal method.

Keywords

Span Tree Minimum Span Tree Cutting Plane Submodular Function Connectivity Constraint 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Andrade, R., Lucena, A., Maculan, N.: Using lagrangian dual information to generate degree constrained spanning trees. Discrete Applied Mathematics 154(5), 703–717 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Arnold, L.R., Bellmore, M.: A bounding minimization problem for primal integer programming. Operations Research 22, 383–392 (1974)zbMATHMathSciNetGoogle Scholar
  3. 3.
    Arnold, L.R., Bellmore, M.: A generated cut for primal integer programming. Operations Research 22, 137–143 (1974)zbMATHMathSciNetGoogle Scholar
  4. 4.
    Arnold, L.R., Bellmore, M.: Iteration skipping in primal integer programming. Operations Research 22, 129–136 (1974)zbMATHMathSciNetGoogle Scholar
  5. 5.
    Barahona, F., Titan, H.: Max mean cuts and max cuts. In: Combinatorial Optimization in Science and Technology, pp. 30–45 (1991)Google Scholar
  6. 6.
    Caccetta, L., Hill, S.P.: A branch and cut method for the degree-constrained minimum spanning tree problem. Networks 37(2), 74–83 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    De Simone, C., Rinaldi, G.: A cutting plane algorithm for the max-cut problem. Optimization Methods and Software 3, 195–214 (1994)Google Scholar
  8. 8.
    Edmonds, J.: Submodular functions, matroids, and certain polyhedra. In: Combinatorial Structures and their Applications, pp. 69–87. Gordon and Breach, New York (1970)Google Scholar
  9. 9.
    Edmonds, J.: Matroids and the greedy algorithm. Math. Programming 1, 127–136 (1971)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Eisenbrand, F., Rinaldi, G., Ventura, P.: 0/1 optimization and 0/1 primal separation are equivalent. In: Proceedings of the 13th annual ACM-SIAM symposium on discrete algorithms, SODA ’02, pp. 920–926 (2002)Google Scholar
  11. 11.
    Garey, M.R., Johnson, D.S.: Computers and Intractability, A Guide to the Theory of NP-Completeness. Freeman, San Francisco (1979)zbMATHGoogle Scholar
  12. 12.
    Glover, F.: A new foundation for a simplified primal integer programming algorithm. Operations Research 16, 727–740 (1968)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Goemans, M.X.: Minimum bounded-degree spanning trees. In: Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science, pp. 273–282. IEEE Computer Society Press, Los Alamitos (2006)Google Scholar
  14. 14.
    Grötschel, M., Lovász, L.: Handbook of Combinatorics, In: Combinatorial Optimization (chapter), vol. 2, pp. 1541–1597. North Holland (1995)Google Scholar
  15. 15.
    Grötschel, M., Lovász, L., Schrijver, A.: The ellipsoid method and its consequences in combinatorial optimization. Combinatorica 1(2), 169–197 (1981)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Karp, R.M., Papadimitriou, C.H.: On linear characterizations of combinatorial optimization problems. In: 21st Annual Symposium on Foundations of Computer Science, Syracuse, New York pp. 1–9 (1980)Google Scholar
  17. 17.
    Knowles, J.D., Corne, D.W.: A new evolutionary approach to the degree-constrained minimum spanning tree problem. IEEE Transactions on Evolutionary Computation 4(2), 125–134 (2000)CrossRefGoogle Scholar
  18. 18.
    Krishnamoorthy, M., Ernst, A.T., Sharaiha, Y.M.: Comparison of algorithms for the degree constrained minimum spanning tree. Journal of Heuristics 7, 587–611 (2001)zbMATHCrossRefGoogle Scholar
  19. 19.
    Kruskal, J.B.: On the shortest spanning subtree of a graph and the traveling salesman problem. Proceedings of the American Mathematics Society 7(1), 48–50 (1956)CrossRefMathSciNetGoogle Scholar
  20. 20.
    Letchford, A.N., Lodi, A.: Primal cutting plane algorithms revisited. Mathematical Methods of Operations Research 56(1), 67–81 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Letchford, A.N., Lodi, A.: An augment-and-branch-and-cut framework for mixed 0-1 programming. In: Jünger, M., Reinelt, G., Rinaldi, G. (eds.) Combinatorial Optimization - Eureka, You Shrink! LNCS, vol. 2570. Springer, Heidelberg (2003)Google Scholar
  22. 22.
    Letchford, A.N., Lodi, A.: Primal separation algorithms. 4OR 1(3), 209–224 (2003)zbMATHMathSciNetGoogle Scholar
  23. 23.
    Nagamochi, H., Ibaraki, T.: Computing edge connectivity in multigraphs and capacitated graphs. SIAM Journal on Discrete Mathematics 5, 54–66 (1992)zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Narula, S.C., Ho, C.A.: Degree-constrained minimum spanning tree. Computers & Operations Research 7, 239–249 (1980)CrossRefGoogle Scholar
  25. 25.
    Padberg, M.W., Grötschel, M.: The Travelling Salesman Problem: A Guided Tour of Combinatorial Optimization. In: Polyhedral computations (chapter), pp. 307–360. Wiley, Chichester (1985)Google Scholar
  26. 26.
    Padberg, M.W., Hong, S.: On the symmetric travelling salesman problem: a computational study. Mathematical Programming Study 12, 78–107 (1980)zbMATHMathSciNetGoogle Scholar
  27. 27.
    Padberg, M.W., Rao, M.R.: The russian method for linear programming III: Bounded integer programming. Technical Report 81-39, Graduate School of Business and Administration, New York University (1981)Google Scholar
  28. 28.
    Padberg, M.W., Wolsey, L.A.: Trees and cuts. Annals of Discrete Mathematics 17, 511–517 (1983)zbMATHMathSciNetGoogle Scholar
  29. 29.
    Prim, R.: Shortest connection networks and some generalizations. Bell System Technical Journal 36, 1389–1401 (1957)Google Scholar
  30. 30.
    Raidl, G.R.: personal communicationGoogle Scholar
  31. 31.
    Raidl, G.R.: An efficient evolutionary algorithm for the degree-constrained minimum spanning tree problem. In: Proceedings of the 2000 IEEE Congress on Evolutionary Computation, vol. 1, pp. 104–111 (2000)Google Scholar
  32. 32.
    Ribeiro, C.C., Souza, M.C.: Variable neighborhood search for the degree-constrained minimum spanning tree problem. Discrete Applied Mathematics 118(1-2), 43–54 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  33. 33.
    Savelsbergh, M., Volgenant, T.: Edge exchanges in the degree-constrained minimum spanning tree problem. Computers & Operations Research 12, 341–348 (1985)zbMATHCrossRefGoogle Scholar
  34. 34.
    Schulz, A.S., Weismantel, R., Ziegler, G.M.: 0/1 integer programming: Optimization and augmentation are equivalent. In: Spirakis, P.G. (ed.) ESA 1995. LNCS, vol. 979, pp. 473–483. Springer, Heidelberg (1995)Google Scholar
  35. 35.
    Sharma, S., Sharma, B.: New technique for solving primal all-integer linear programming. Opsearch 34, 62–68 (1997)zbMATHMathSciNetGoogle Scholar
  36. 36.
    Volgenant, A.: A lagrangean approach to the degree-constrained minimum spanning tree problem. European Journal of Operational Research 39, 325–331 (1989)zbMATHCrossRefMathSciNetGoogle Scholar
  37. 37.
    Wolsey, L.A.: Integer Programming. Wiley-Interscience, New York, USA (1998)zbMATHGoogle Scholar
  38. 38.
    Young, R.D.: A simplified primal (all-integer) integer programming algorithm. Operations Research 16, 750–782 (1968)zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Berlin Heidelberg 2007

Authors and Affiliations

  • Markus Behle
    • 1
  • Michael Jünger
    • 2
  • Frauke Liers
    • 2
  1. 1.Max-Planck-Institut für Informatik, Stuhlsatzenhausweg 85, 66123 SaarbrückenGermany
  2. 2.Universität zu Köln, Institut für Informatik, Pohligstrasse 1, 50969 KölnGermany

Personalised recommendations