Lower and upper bounds for the degree-constrained minimum spanning tree problem

Networks

Volumn 50, Issue 1, 2007, Pages 55-66

  Da Cunha, Alexandre Salles ^a   Lucena, Abílio ^b  

^a FEDERAL UNIVERSITY OF MINAS GERAIS   (Brazil)

^b FEDERAL UNIVERSITY OF RIO DE JANEIRO   (Brazil)

Author keywords

Blossom inequalities; Degree constrained minimum spanning tree; Lagrangian relaxation; Lagrangian warm start to cutting plane algorithms; Non delayed relax and cut

Indexed keywords

ALGORITHMS; DECISION TREES; LINEAR PROGRAMMING; OPTIMIZATION;

BLOSSOM INEQUALITIES; KRUSKAL STYLE ALGORITHM; LAGRANGIAN INFORMATION; SPANNING TREE PROBLEM;

CONSTRAINT THEORY;

EID: 34547804399     PISSN: 00283045     EISSN: 10970037     Source Type: Journal    
DOI: 10.1002/net.20166     Document Type: Article

References (52)

1. R. Andrade, A. Lucena, and N. Maculan, Using lagrangian dual information to generate degree constrained minimum spanning trees, Discrete Appl Math 154 (2006), 707-717.
2. A. Belloni and A. Lucena, A relax and cut algorithm for the traveling salesman problem, 17th International Symposium on Mathematical Programming, Atlanta, USA, 2000, pp. 77.
3. A. Belloni and A. Lucena, "A Lagrangian heuristic for the linear ordering problem", Metaheuristics: Computer decisionmaking, M.G.C. Resende and J. Pinho de Sousa (Editors), Kluwer Academic, Boston, 2003, pp. 123-151.
4. L. Caccetta and S.P. Hill, A branch and cut method for the degree-constrained minimum spanning tree problem, Networks 37 (2001), 74-83.
5. F. Calheiros, A. Lucena, and C. de Souza, Optimal rectangular partitions, Networks 41 (2003), 51-67.
6. G. Craig, M. Krishnamoorthy, and M. Palaniswami, "Comparison of heuristic algorithms for the degree constrained minimum spanning trees", Metaheuristics: Theory and applications, I.H. Osman and J.P. Kelly (Editors), Kluwer Academic, Boston, 1996, pp. 83-96.
7. A.S. Cunha and A. Lucena, Lower and upper bounds for the degree-constrained minimum spanning tree problem, INOC 2005 Proceedings, Vol. 1, Lisbon, Portugal, 2005, pp. 186-194.
8. C. Duin, Steiner's problem in graphs, Ph.D. Thesis, University of Amsterdam, 1993.
9. J. Edmonds, Matroids and the greedy algorithm, Math Programming 1 (1971), 127-136.
10. J. Edmonds and E.L. Johnson, Matching: A well-solved class of integer linear programs, Proc Calgary Int Conf Combinatorial Struct their Appl, R.K. Guy et al. (Editor), Gordon and Breach, New York, 1970, pp. 89-92.
11. L. Escudero, M. Guignard, and K. Malik, A Lagrangian relax and cut approach for the sequential ordering with precedence constraints, Ann Oper Res 50 (1994), 219-237.
12. H.N. Gabow, Two algorithms for generating weighted spanning trees in order, SIAM J Comput 6 (1977), 139-150.
13. H.N. Gabow, A good algorithm for smallest trees with a degree constraint, Networks 8 (1978), 201-208.
14. A.B. Gamble and W.R. Pulleyblank, Forest covers and a polyhedral intersection theorem, Math Programming 45 (1989), 49-58.
15. M.R. Garey and D.S. Johnson, Computers and intractability: A guide to the theory of NP-completeness, Freeman, San Francisco, 1979.
16. B. Gavish, Topological design of centralized computer networks-formulations and algorithms, Networks 12 (1982), 355-377.
17. F. Glover and D. Klingman, "Finding minimum spanning trees with a fixed number of links at a node", Combinatorial programming: Methods and applications, B. Roy (Editor), Reidell, Dordrecht, 1975, pp. 191-201.
18. F. Glover and D. Klingman, Recent developments in computer implementation technology for network flow algorithms, INFOR 20 (1982), 433-452.
19. M. Grötschel and M. Padberg, On the symmetric traveling salesman problem I: Inequalities, Math Programming 16 (1979), 265-280.
20. M. Grötschel and W.R. Pulleyblank, Clique tree inequalities and the symmetric traveling salesman problem, Math Oper Res 11 (1986), 537-569.
21. L. Hall and T.L. Magnanti, A polyhedral intersection theorem for capacitated spanning trees, Math Oper Res 17 (1982), 398-410.
22. M. Held and R.M. Karp, The traveling salesman problem and minimum spanning trees, Oper Res 18 (1970), 1138-1162.
23. M. Held and R.M. Karp, The traveling salesman problem and minimum spanning trees: Part II, Math Programming 1 (1971), 6-25.
24. M.H. Held, P. Wolfe, and H.D. Crowder, Validation of subgradient optimization, Math Programming 6 (1974), 62-88.
25. M. Hunting, U. Faigle, and W. Kern, A Lagrangian relaxation approach to the edge-weighted clique problem, Eur J Oper Res 131 (2001), 119-131.
26. N. Katoh, T. Ibaraki, and H. Mine, An algorithm for finding K minimum spanning trees, SIAM J Comput 10 (1981), 247-255.
27. S. Khuller, B. Raghavachari, and N. Young, Low degree spanning trees of small weight, SIAM J Comput 25 (1996), 355-368.
28. J. Knowles and D. Corne, A new evolutionary approach to the degree constrained minimum spanning tree problem, IEEE Trans Evol Comput 4 (2000), 125-134.
29. M. Krishnamoorthy, T.A. Ernst, and Y.M. Sharaiha, Comparison of algorithms for the degree constrained minimum spanning tree, J Heuristics 7 (2001), 587-611.
30. J.B. Kruskal, On the shortest spanning tree of a graph and the traveling salesman problem, Proc Am Math Soc 7 (1956), 48-50.
31. A.N. Letchford, G. Reinelt, and D.O. Theis, A faster exact separation algorithm for blossom inequalities, Lect Notes Comput Sci 3064 (2004), 196-205.
32. A. Lucena, Steiner problem in graphs: Lagrangean relaxation and cutting planes, COAL Bull 21 (1992), 2-8.
33. A. Lucena, Tight bounds for the Steiner problem in graphs, Proc NETFLOW 93, Tech Report TR-21/93, Dipartimento di Informatica, Univesitá degli Studi di Pisa, Pisa, Italy, 1993, pp. 147-154.
34. A. Lucena, Non delayed relax-and-cut algorithms, Ann Oper Res 140 (2005), 375-410.
35. T. Magnanti and L. Wolsey, "Optimal trees", Handbooks in operations research and management science, T. Magnanti, M. Ball, C. Monma, and G. Nemhauser (Editors), Vol. 6, North-Holland, Amsterdam, 1995, pp. 503-615.
36. C. Martinhon, A. Lucena, and N. Maculan, Stronger K-tree relaxations for the vehicle routing problem, Eur J Oper Res 158 (2004), 56-71.
37. S.C. Narula and C.A. Ho, Degree-constrained minimum spanning tree, Comput Oper Res 7 (1980), 239-249.
38. G. Nemhauser and L. Wolsey, Integer and combinatorial optimization, Wiley, New York, 1988.
39. M. Padberg and M.R. Rao, Odd minimum cut-sets and b-matchings, Math Oper Res 7 (1982), 67-80.
40. M. Padberg and L. Wolsey, Trees and cuts, Ann Discrete Math 17 (1983), 511-517.
41. M. de Moraes Palmeira, A. Lucena, and O. Porto, A relax and cut algorithm for the quadratic knapsack problem, 17th Int Symp Math Programming, Atlanta, USA, 2000, pp. 102.
42. G.L. Nemhauser and G. Sigismondi, A strong cutting plane branch-and-bound algorithm for node packing, J Oper Res Soc 43 (1992), 443-157.
43. C. Papadimitriou and U.V. Vazirani, On two geometric problems related to the traveling salesman problem, J Algorithms 5 (1984), 231-246.
44. R.C. Prim, Shortest connection networks and some generalizations, Bell Syst Tech J 36 (1957), 1389-1401.
45. C.C. Ribeiro and M.C. Souza, Tabu search for the Steiner problem in graphs, Networks 36 (2000), 138-146.
46. C.C Ribeiro and M.C. Souza, Variable neighborhood search for the degree-constrained minimum spanning tree problem, Discrete Appl Math 118 (2002), 43-54.
47. M. Savelsbergh and A. Volgenant, Edge exchanges in the degree constrained minimum spanning tree problem, Comput Oper Res 12 (1985), 341-348.
48. J.B.C. da Silva, A Lagrangian heuristic for the capacitated minimum spanning tree problem, M.Sc. Dissertation, Programa de Engenharia de Sistemas e Computaçāo, COPPE, Universidade Federal do Rio de Janeiro, 2002 (in Portuguese).
49. A. Volgenant, A Lagrangean approach to the degreeconstrained minimum spanning tree problem, Eur J Oper Res 39(1989), 325-331.
50. A. Volgenant and R. Jonker, The symmetric traveling salesman problem and edge exchanges in minimal 1-trees, Eur J Oper Res 12 (1983), 394-403.
51. Y. Yamamoto, The Held-Karp algorithm and degreeconstrained minimum 1 -trees, Math Programming 15 (1978), 228-231.
52. G. Zhou and M. Gen, A note on genetic algorithms for degree constrained spanning tree problems, Networks 30 (1997), 91-95.