Computational Optimization and Applications

, Volume 51, Issue 3, pp 1211–1229 | Cite as

A combinatorial optimization algorithm for solving the branchwidth problem

  • J. Cole Smith
  • Elif Ulusal
  • Illya V. Hicks


In this paper, we consider the problem of computing an optimal branch decomposition of a graph. Branch decompositions and branchwidth were introduced by Robertson and Seymour in their series of papers that proved the Graph Minors Theorem. Branch decompositions have proven to be useful in solving many NP-hard problems, such as the traveling salesman, independent set, and ring routing problems, by means of combinatorial algorithms that operate on branch decompositions. We develop an implicit enumeration algorithm for the optimal branch decomposition problem and examine its performance on a set of classical graph instances.


Branch decomposition Branchwidth Implicit enumeration Partitioning 


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  1. 1.
    Arnborg, S., Lagergren, J., Seese, D.: Easy problems for tree-decomposable graphs. J. Algorithms 12, 308–340 (1991) MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Bian, Z., Gu, Q., Marzban, M., Tamaki, H., Yoshitake, Y.: Empirical study on branchwidth and branch decomposition of planar graphs. In: Proceedings of the 2008 SIAM Workshop on Algorithm Engineering and Experiments (ALENEX), pp. 152–165. SIAM, Philadelphia (2008) Google Scholar
  3. 3.
    Bodlaender, H.L., Fomin, F.V., Koster, A.M.C.A., Kratsch, D., Thilikos, D.M.: On exact algorithms for treewidth. In: Proceedings of the 14th Annual European Symposium on Algorithms (ESA 2006). Lecture Notes in Computer Science, vol. 4168, pp. 672–683. Springer, Berlin (2006) Google Scholar
  4. 4.
    Bodlaender, H.L., Koster, A.M.C.A., Wolle, T.: Contraction and treewidth lower bounds. J. Graph Algorithms Appl. 10(1), 5–49 (2006) MathSciNetMATHGoogle Scholar
  5. 5.
    Cook, W.J., Seymour, P.D.: An algorithm for the ring-router problem. Technical report, Bellcore (1994) Google Scholar
  6. 6.
    Cook, W.J., Seymour, P.D.: Tour merging via branch-decomposition. INFORMS J. Comput. 15(3), 233–248 (2003) MathSciNetCrossRefGoogle Scholar
  7. 7.
    Courcelle, B.: The monadic second-order logic of graphs I: recognizable set of finite graphs. Inf. Comput. 85, 12–75 (1990) MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Fomin, F.V., Thilikos, D.M.: A simple and fast approach for solving problems on planar graphs. In: Lecture Notes in Computer Science, vol. 2996, pp. 56–67. Springer, Berlin (2004) Google Scholar
  9. 9.
    Fomin, F.V., Fraigniaud, P., Thilikos, D.M.: The price of connectedness in expansions. Technical Report 273, Department of Informatics, University of Bergen, Bergen, Norway, May 2004 Google Scholar
  10. 10.
    Fomin, F.V., Mazoit, F., Todinca, I.: Computing branchwidth via efficient triangulations and blocks. Discrete Appl. Math. 157(12), 2726–2736 (2009) MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Garey, M.R., Graham, R.L., Johnson, D.S., Knuth, D.E.: Complexity results for bandwidth minimization. SIAM J. Appl. Math. 34, 477–495 (1978) MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Hicks, I.V.: Branchwidth heuristics. Congr. Numer. 159, 31–50 (2002) MathSciNetMATHGoogle Scholar
  13. 13.
    Hicks, I.V.: Branch decompositions and minor containment. Networks 43(1), 1–9 (2004) MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Hicks, I.V.: Graphs, branchwidth, and tangles! Oh my! Networks 45(2), 55–60 (2005) MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Hicks, I.V.: Planar branch decompositions II: the cycle method. INFORMS J. Comput. 17(4), 413–421 (2005) MathSciNetCrossRefGoogle Scholar
  16. 16.
    Monien, B.: The bandwidth minimization problem for caterpillars with hair length 3 is NP-Complete. SIAM J. Algebr. Discrete Methods 7, 505–512 (1986) MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Robertson, N., Seymour, P.: Graph minors. X. Obstructions to tree-decomposition. J. Comb. Theory, Ser. B 52, 153–190 (1991) MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Robertson, N., Seymour, P.D.: Graph minors. XIII. The disjoint paths problem. J. Comb. Theory, Ser. B 63, 65–110 (1995) MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Seymour, P.D., Thomas, R.: Call routing and the ratcatcher. Combinatorica 14(2), 217–241 (1994) MathSciNetMATHCrossRefGoogle Scholar

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© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Industrial & Systems EngineeringUniversity of FloridaGainesvilleUSA
  2. 2.MathematicsHouston Community CollegeHoustonUSA
  3. 3.Computational & Applied MathematicsRice UniversityHoustonUSA

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