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A combinatorial optimization algorithm for solving the branchwidth problem

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Abstract

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.

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References

  1. Arnborg, S., Lagergren, J., Seese, D.: Easy problems for tree-decomposable graphs. J. Algorithms 12, 308–340 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  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. 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. Bodlaender, H.L., Koster, A.M.C.A., Wolle, T.: Contraction and treewidth lower bounds. J. Graph Algorithms Appl. 10(1), 5–49 (2006)

    MathSciNet  MATH  Google Scholar 

  5. Cook, W.J., Seymour, P.D.: An algorithm for the ring-router problem. Technical report, Bellcore (1994)

  6. Cook, W.J., Seymour, P.D.: Tour merging via branch-decomposition. INFORMS J. Comput. 15(3), 233–248 (2003)

    Article  MathSciNet  Google Scholar 

  7. Courcelle, B.: The monadic second-order logic of graphs I: recognizable set of finite graphs. Inf. Comput. 85, 12–75 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  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. 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

  10. Fomin, F.V., Mazoit, F., Todinca, I.: Computing branchwidth via efficient triangulations and blocks. Discrete Appl. Math. 157(12), 2726–2736 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  12. Hicks, I.V.: Branchwidth heuristics. Congr. Numer. 159, 31–50 (2002)

    MathSciNet  MATH  Google Scholar 

  13. Hicks, I.V.: Branch decompositions and minor containment. Networks 43(1), 1–9 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  14. Hicks, I.V.: Graphs, branchwidth, and tangles! Oh my! Networks 45(2), 55–60 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  15. Hicks, I.V.: Planar branch decompositions II: the cycle method. INFORMS J. Comput. 17(4), 413–421 (2005)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  17. Robertson, N., Seymour, P.: Graph minors. X. Obstructions to tree-decomposition. J. Comb. Theory, Ser. B 52, 153–190 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  18. Robertson, N., Seymour, P.D.: Graph minors. XIII. The disjoint paths problem. J. Comb. Theory, Ser. B 63, 65–110 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  19. Seymour, P.D., Thomas, R.: Call routing and the ratcatcher. Combinatorica 14(2), 217–241 (1994)

    Article  MathSciNet  MATH  Google Scholar 

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Correspondence to J. Cole Smith.

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Smith, J.C., Ulusal, E. & Hicks, I.V. A combinatorial optimization algorithm for solving the branchwidth problem. Comput Optim Appl 51, 1211–1229 (2012). https://doi.org/10.1007/s10589-011-9397-z

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