Abstract
In this paper we present a new technique for computing lower bounds for graph treewidth. Our technique is based on the characterisation of the treewidth as the maximum order of a bramble of the graph. We give two algorithms: one for general graphs, and one for planar graphs. The algorithm for planar graphs is shown to give a lower bound for the treewidth that is at most a constant factor away from the exact treewidth. For both algorithms, we report on extensive computational experiments that show that the algorithms give often excellent lower bounds, in particular when applied to (close to) planar graphs.
This work was partially supported by the Netherlands Organisation for Scientific Research NWO (project Treewidth and Combinatorial Optimisation) and partially by the DFG research group ”Algorithms, Structure, Randomness” (Grant number GR 883/9-3, GR 883/9-4).
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Ahuja, R.K., Magnanti, T.L., Orlin, J.B.: Network Flows: Theory, Algorithms, and Applications. Prentice-Hall, Englewood Cliffs (1993)
Amir, E.: Efficient approximations for triangulation of minimum treewidth. In: Proceedings of the 17th Conference on Uncertainty in Artificial Intelligence, pp. 7–15 (2001)
Arnborg, S., Corneil, D.G., Proskurowski, A.: Complexity of finding embeddings in a k-tree. SIAM J. Alg. Disc. Meth. 8, 277–284 (1987)
Bellenbaum, P., Diestel, R.: Two short proofs concerning tree-decompositions. Combinatorics, Probability, and Computing 11, 541–547 (2002)
Bodlaender, H.L.: A linear time algorithm for finding tree-decompositions of small treewidth. SIAM J. Comput. 25, 1305–1317 (1996)
Bodlaender, H.L.: Discovering treewidth. In: Vojtáš, P., Bieliková, M., Charron-Bost, B., Sýkora, O. (eds.) SOFSEM 2005. LNCS, vol. 3381, pp. 1–16. Springer, Heidelberg (2005)
Bodlaender, H.L., Koster, A.M.C.A.: On the Maximum Cardinality Search lower bound for treewidth. In: Hromkovič, J., Nagl, M., Westfechtel, B. (eds.) WG 2004. LNCS, vol. 3353, pp. 81–92. Springer, Heidelberg (2004)
Bodlaender, H.L., Koster, A.M.C.A.: Safe separators for treewidth. In: Proceedings 6th Workshop on Algorithm Engineering and Experiments, ALENEX 2004, pp. 70–78 (2004)
Bodlaender, H.L., Koster, A.M.C.A., van de Eijkhof, F., van der Gaag, L.C.: Pre-processing for triangulation of probabilistic networks. In: Breese, J., Koller, D. (eds.) Proceedings of the 17th Conference on Uncertainty in Artificial Intelligence, pp. 32–39. Morgan Kaufmann, San Francisco (2001)
Bodlaender, H.L., Koster, A.M.C.A., Wolle, T.: Contraction and treewidth lower bounds. In: Albers, S., Radzik, T. (eds.) ESA 2004. LNCS, vol. 3221, pp. 628–639. Springer, Heidelberg (2004)
Bouchitté, V., Todinca, I.: Treewidth and minimum fill-in: Grouping the minimal separators. SIAM J. Comput. 31, 212–232 (2001)
Clautiaux, F., Carlier, J., Moukrim, A., Négre, S.: New lower and upper bounds for graph treewidth. In: Rolim, J.D.P. (ed.) WEA 2003. LNCS, vol. 2647, pp. 70–80. Springer, Heidelberg (2003)
Clautiaux, F., Moukrim, A., Négre, S., Carlier, J.: Heuristic and meta-heuristic methods for computing graph treewidth. RAIRO Oper. Res. 38, 13–26 (2004)
Cook, W., Seymour, P.D.: Tour merging via branch-decomposition. Informs J. on Computing 15(3), 233–248 (2003)
Demaine, E.D., Hajiaghayi, M.: Graphs excluding a fixed minor have grids as large as treewidth, with combinatorial and algorithmic applications through bidimensionality. In: Proceedings of the 16th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2005, pp. 682–689 (2005)
Diestel, R., Jensen, T.R., Gorbunov, K.Y., Thomassen, C.: Highly connected sets and the excluded grid theorem. J. Comb. Theory Series B 75, 61–73 (1999)
Fomin, F.V., Kratsch, D., Todinca, I.: Exact (exponential) algorithms for treewidth and minimum fill-in. In: Díaz, J., Karhumäki, J., Lepistö, A., Sannella, D. (eds.) ICALP 2004. LNCS, vol. 3142, pp. 568–580. Springer, Heidelberg (2004)
Gogate, V., Dechter, R.: A complete anytime algorithm for treewidth. In: Proceedings of the 20th Annual Conference on Uncertainty in Artificial Intelligence UAI 2004, pp. 201–208. AUAI Press, Arlington (2004)
Hicks, I.V.: Planar branch decompositions I: The ratcatcher. INFORMS Journal on Computing (2005) (to appear)
Hicks, I.V.: Planar branch decompositions II: The cycle method. INFORMS Journal on Computing (2005) (to appear)
Koster, A.M.C.A., Bodlaender, H.L., van Hoesel, S.P.M.: Treewidth: Computational experiments. In: Broersma, H., Faigle, U., Hurink, J., Pickl, S. (eds.) Electronic Notes in Discrete Mathematics, vol. 8. Elsevier Science Publishers, Amsterdam (2001)
Koster, A.M.C.A., Wolle, T., Bodlaender, H.L.: Degree-based treewidth lower bounds. In: Nikoletseas, S.E. (ed.) WEA 2005. LNCS, vol. 3503, pp. 101–112. Springer, Heidelberg (2005)
Lucena, B.: A new lower bound for tree-width using maximum cardinality search. SIAM J. Disc. Math. 16, 345–353 (2003)
Ramachandramurthi, S.: The structure and number of obstructions to treewidth. SIAM J. Disc. Math. 10, 146–157 (1997)
Reed, B.A.: Tree width and tangles, a new measure of connectivity and some applications. LMS Lecture Note Series, vol. 241, pp. 87–162. Cambridge University Press, Cambridge (1997)
Robertson, N., Seymour, P.D., Thomas, R.: Quickly excluding a planar graph. J. Comb. Theory Series B 62, 323–348 (1994)
Seymour, P.D., Thomas, R.: Graph searching and a minimax theorem for tree-width. J. Comb. Theory Series B 58, 239–257 (1993)
Seymour, P.D., Thomas, R.: Call routing and the ratcatcher. Combinatorica 14(2), 217–241 (1994)
West, D.B.: Introduction to graph theory. Prentice-Hall, Englewood Cliffs (2001)
Wolle, T., Koster, A.M.C.A., Bodlaender, H.L.: A note on contraction degeneracy. Technical Report UU-CS-2004-042, Institute of Information and Computing Sciences, Utrecht University, Utrecht, The Netherlands (2004)
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Bodlaender, H.L., Grigoriev, A., Koster, A.M.C.A. (2005). Treewidth Lower Bounds with Brambles. In: Brodal, G.S., Leonardi, S. (eds) Algorithms – ESA 2005. ESA 2005. Lecture Notes in Computer Science, vol 3669. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11561071_36
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DOI: https://doi.org/10.1007/11561071_36
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