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

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Experimental Algorithms (WEA 2007)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 4525))

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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.

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References

  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)

    Article  MATH  MathSciNet  Google Scholar 

  2. Arnold, L.R., Bellmore, M.: A bounding minimization problem for primal integer programming. Operations Research 22, 383–392 (1974)

    MATH  MathSciNet  Google Scholar 

  3. Arnold, L.R., Bellmore, M.: A generated cut for primal integer programming. Operations Research 22, 137–143 (1974)

    MATH  MathSciNet  Google Scholar 

  4. Arnold, L.R., Bellmore, M.: Iteration skipping in primal integer programming. Operations Research 22, 129–136 (1974)

    MATH  MathSciNet  Google Scholar 

  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. Caccetta, L., Hill, S.P.: A branch and cut method for the degree-constrained minimum spanning tree problem. Networks 37(2), 74–83 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  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. 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. Edmonds, J.: Matroids and the greedy algorithm. Math. Programming 1, 127–136 (1971)

    Article  MATH  MathSciNet  Google Scholar 

  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. Garey, M.R., Johnson, D.S.: Computers and Intractability, A Guide to the Theory of NP-Completeness. Freeman, San Francisco (1979)

    MATH  Google Scholar 

  12. Glover, F.: A new foundation for a simplified primal integer programming algorithm. Operations Research 16, 727–740 (1968)

    Article  MathSciNet  Google Scholar 

  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. Grötschel, M., Lovász, L.: Handbook of Combinatorics, In: Combinatorial Optimization (chapter), vol. 2, pp. 1541–1597. North Holland (1995)

    Google Scholar 

  15. Grötschel, M., Lovász, L., Schrijver, A.: The ellipsoid method and its consequences in combinatorial optimization. Combinatorica 1(2), 169–197 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  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. 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)

    Article  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  20. Letchford, A.N., Lodi, A.: Primal cutting plane algorithms revisited. Mathematical Methods of Operations Research 56(1), 67–81 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  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. Letchford, A.N., Lodi, A.: Primal separation algorithms. 4OR 1(3), 209–224 (2003)

    MATH  MathSciNet  Google Scholar 

  23. Nagamochi, H., Ibaraki, T.: Computing edge connectivity in multigraphs and capacitated graphs. SIAM Journal on Discrete Mathematics 5, 54–66 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  24. Narula, S.C., Ho, C.A.: Degree-constrained minimum spanning tree. Computers & Operations Research 7, 239–249 (1980)

    Article  Google Scholar 

  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. Padberg, M.W., Hong, S.: On the symmetric travelling salesman problem: a computational study. Mathematical Programming Study 12, 78–107 (1980)

    MATH  MathSciNet  Google Scholar 

  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. Padberg, M.W., Wolsey, L.A.: Trees and cuts. Annals of Discrete Mathematics 17, 511–517 (1983)

    MATH  MathSciNet  Google Scholar 

  29. Prim, R.: Shortest connection networks and some generalizations. Bell System Technical Journal 36, 1389–1401 (1957)

    Google Scholar 

  30. Raidl, G.R.: personal communication

    Google Scholar 

  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. 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)

    Article  MATH  MathSciNet  Google Scholar 

  33. Savelsbergh, M., Volgenant, T.: Edge exchanges in the degree-constrained minimum spanning tree problem. Computers & Operations Research 12, 341–348 (1985)

    Article  MATH  Google Scholar 

  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. Sharma, S., Sharma, B.: New technique for solving primal all-integer linear programming. Opsearch 34, 62–68 (1997)

    MATH  MathSciNet  Google Scholar 

  36. Volgenant, A.: A lagrangean approach to the degree-constrained minimum spanning tree problem. European Journal of Operational Research 39, 325–331 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  37. Wolsey, L.A.: Integer Programming. Wiley-Interscience, New York, USA (1998)

    MATH  Google Scholar 

  38. Young, R.D.: A simplified primal (all-integer) integer programming algorithm. Operations Research 16, 750–782 (1968)

    MATH  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Max-Planck-Institut für Informatik, Stuhlsatzenhausweg 85, 66123 Saarbrücken, Germany

    Markus Behle

  2. Universität zu Köln, Institut für Informatik, Pohligstrasse 1, 50969 Köln, Germany

    Michael Jünger & Frauke Liers

Authors
  1. Markus Behle
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  2. Michael Jünger
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  3. Frauke Liers
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Camil Demetrescu

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Behle, M., Jünger, M., Liers, F. (2007). A Primal Branch-and-Cut Algorithm for the Degree-Constrained Minimum Spanning Tree Problem. In: Demetrescu, C. (eds) Experimental Algorithms. WEA 2007. Lecture Notes in Computer Science, vol 4525. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-72845-0_29

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  • DOI: https://doi.org/10.1007/978-3-540-72845-0_29

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-72844-3

  • Online ISBN: 978-3-540-72845-0

  • eBook Packages: Computer ScienceComputer Science (R0)

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