Abstract
Every lower bound for treewidth can be extended by taking the maximum of the lower bound over all subgraphs or minors. This extension is shown to be a very vital idea for improving treewidth lower bounds. In this paper, we investigate a total of nine graph parameters, providing lower bounds for treewidth. The parameters have in common that they all are the vertex-degree of some vertex in a subgraph or minor of the input graph. We show relations between these graph parameters and study their computational complexity. To allow a practical comparison of the bounds, we developed heuristic algorithms for those parameters that are N P-hard to compute. Computational experiments show that combining the treewidth lower bounds with minors can considerably improve the lower bounds.
This work was partially supported by the DFG research group ”Algorithms, Structure, Randomness” (Grant number GR 883/9-3, GR 883/9-4), and partially by the Netherlands Organisation for Scientific Research NWO (project Treewidth and Combinatorial Optimisation).
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Arnborg, S., Corneil, D.G., Proskurowski, A.: Complexity of finding embeddings in a k-tree. SIAM J. Alg. Disc. Meth. 8, 277–284 (1987)
Behzad, M., Chartrand, G., Lesniak-Foster, L.: Graphs and Digraphs. Pindle, Weber & Schmidt, Boston (1979)
Bodlaender, H.L.: A partial k-arboretum of graphs with bounded treewidth. Theor. Comp. Sc. 209, 1–45 (1998)
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 ALENEX2004, pp. 70–78 (2004)
Bodlaender, H.L., Koster, A.M.C.A., Eijkhof, F.v.d., 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)
Bodlaender, H.L., Koster, A.M.C.A., Wolle, T.: Contraction and treewidth lower bounds. Technical Report UU-CS-2004-34, Dept. of Computer Science, Utrecht University, Utrecht, The Netherlands (2004)
Clautiaux, F., Moukrim, S.N.A., Carlier, J.: Heuristic and meta-heuristic methods for computing graph treewidth. RAIRO Oper. Res. 38, 13–26 (2004)
Clautiaux, F., Carlier, J., Moukrim, A., Négre, S.: New lower and upper bounds for graph treewidth. In: Jansen, K., Margraf, M., Mastrolli, M., Rolim, J.D.P. (eds.) WEA 2003. LNCS, vol. 2647, pp. 70–80. Springer, Heidelberg (2003)
Eijkhof, F.v.d., Bodlaender, H.L.: Safe reduction rules for weighted treewidth. In: Kučera, L. (ed.) WG 2002. LNCS, vol. 2573, pp. 176–185. Springer, Heidelberg (2002)
Gogate, V., Dechter, R.: A complete anytime algorithm for treewidth. In: proceedings UAI 2004. Uncertainty in Artificial Intelligence (2004)
Hicks, I.V.: Planar branch decompositions I: The ratcatcher. 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., van Hoesel, S.P.M., Kolen, A.W.J.: Solving partial constraint satisfaction problems with tree decomposition. Networks 40, 170–180 (2002)
Koster, A.M.C.A., Wolle, T., Bodlaender, H.L.: Degree-based treewidth lower bounds. Technical Report UU-CS-2004-050, Institute for Information and Computing Sciences, Utrecht University, Utrecht, The Netherlands (2004)
Lauritzen, S.J., Spiegelhalter, D.J.: Local computations with probabilities on graphical structures and their application to expert systems. The Journal of the Royal Statistical Society. Series B (Methodological) 50, 157–224 (1988)
Lucena, B.: A new lower bound for tree-width using maximum cardinality search. SIAM J. Disc. Math. 16, 345–353 (2003)
Ramachandramurthi, S.: Algorithms for VLSI Layout Based on Graph Width Metrics. PhD thesis, Computer Science Department, University of Tennessee, Knoxville, Tennessee, USA (1994)
Ramachandramurthi, S.: The structure and number of obstructions to treewidth. SIAM J. Disc. Math. 10, 146–157 (1997)
Robertson, N., Seymour, P.D.: Graph minors. II. Algorithmic aspects of tree-width. J. Algorithms 7, 309–322 (1986)
Seymour, P.D., Thomas, R.: Call routing and the ratcatcher. Combinatorica 14(2), 217–241 (1994)
Treewidthlib. 2004-03-31, http://www.cs.uu.nl/people/hansb/treewidthlib
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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Koster, A.M.C.A., Wolle, T., Bodlaender, H.L. (2005). Degree-Based Treewidth Lower Bounds. In: Nikoletseas, S.E. (eds) Experimental and Efficient Algorithms. WEA 2005. Lecture Notes in Computer Science, vol 3503. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11427186_11
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DOI: https://doi.org/10.1007/11427186_11
Publisher Name: Springer, Berlin, Heidelberg
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