Abstract
It is well-known that for graphs with high edge density the tree-width is always high while the clique-width can be low. Boolean-width is a new parameter that is never higher than tree-width or clique-width and can in fact be as small as logarithmic in clique-width. Boolean-width is defined using a decomposition tree by evaluating the number of neighborhoods across the resulting cuts of the graph. Several NP-hard problems can be solved efficiently by dynamic programming when given a decomposition of boolean-width k, e.g. Max Weight Independent Set in time O(n 2 k22k) and Min Weight Dominating Set in time O(n 2 + nk23k). Finding decompositions of low boolean-width is therefore of practical interest. There is evidence that computing boolean-width is hard, while the existence of a useful approximation algorithm is still open. In this paper we introduce and study a heuristic algorithm that finds a reasonably good decomposition to be used for dynamic programming based on boolean-width. On a set of graphs of practical relevance, specifically graphs in TreewidthLIB, the best known upper bound on their tree-width is compared to the upper bound on their boolean-width given by our heuristic. For the large majority of the graphs on which we made the tests, the tree-width bound is at least twice as big as the boolean-width bound, and boolean-width compares better the higher the edge density. This means that, for problems like Dominating Set, using boolean-width should outperform dynamic programming by tree-width, at least for graphs of edge density above a certain bound. In view of the amount of previous work on heuristics for tree-width these results indicate that boolean-width could in the future outperform tree-width in practice for a large class of graphs and problems.
Supported by the Norwegian Research Council, project PARALGO.
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References
Adler, I., Bui-Xuan, B.M., Rabinovich, Y., Renault, G., Telle, J.A., Vatshelle, M.: On the Boolean-Width of a Graph: Structure and Applications. In: Thilikos, D.M. (ed.) WG 2010. LNCS, vol. 6410, pp. 159–170. Springer, Heidelberg (2010)
Belmonte, R., Vatshelle, M.: Graph classes with structured neighborhoods and algorithmic applications. In: Proceedings of the 37th International Workshop on Graph-Theoretic Concepts in Computer Science, WG 2011 (2011), www.ii.uib.no/~martinv/Papers/LogBoolw.pdf
Bodlaender, H.L.: A linear time algorithm for finding tree-decompositions of small treewidth. SIAM Journal on Computing 25, 1305–1317 (1996)
Bodlaender, H.L.: Treewidth: Characterizations, Applications, and Computations. In: Fomin, F.V. (ed.) WG 2006. LNCS, vol. 4271, pp. 1–14. Springer, Heidelberg (2006)
Bodlaender, H.L., Koster, A.M.C.A.: Treewidth computations I. Upper bounds. Information and Computation 208, 259–275 (2010)
Bodlaender, H.L., Koster, A.M.C.A.: Treewidth computations II. lower bounds. Technical Report UU-CS-2010-022, Department of Information and Computing Sciences, Utrecht University, Utrecht, The Netherlands (2010) (accepted for publication in Information and Computation)
Brandstadt, A.: Personal Communication
Bui-Xuan, B.M., Telle, J.A., Vatshelle, M.: Boolean-width of graphs. Theoretical Computer Science (to appear, 2011), www.ii.uib.no/~telle/bib/listofpub/BTV11.pdf
Chen, H.: Quantified constraint satisfaction and bounded treewidth. In: de Mántaras, R.L., Saitta, L. (eds.) Proceedings of the 17th European Conference on Artificial Intelligence, ECAI 2004, pp. 161–165 (2004)
The second DIMACS implementation challenge: NP-Hard Problems: Maximum Clique, Graph Coloring, and Satisfiability (1992-1993), http://dimacs.rutgers.edu/Challenges/
Gottlob, G., Leone, N., Scarcello, F.: A comparison of structural CSP decomposition methods. Acta Informatica 124, 243–282 (2000)
Hicks, I.V., Koster, A.M.C.A., Kolotoğlu, E.: Branch and tree decomposition techniques for discrete optimization. In: Cole Smith, J. (ed.) INFORMS Annual Meeting, TutORials 2005. INFORMS Tutorials in Operations Research Series, ch. 1, pp. 1–29 (2005)
Hliněný, P., Oum, S.: Finding branch-decomposition and rank-decomposition. SIAM Journal on Computing 38, 1012–1032 (2008)
Kim, K.H.: Boolean matrix theory and its applications. Marcel Dekker (1982)
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)
Overwijk, A., Penninkx, E., Bodlaender, H.L.: A Local Search Algorithm for Branchwidth. In: Černá, I., Gyimóthy, T., Hromkovič, J., Jefferey, K., Králović, R., Vukolić, M., Wolf, S. (eds.) SOFSEM 2011. LNCS, vol. 6543, pp. 444–454. Springer, Heidelberg (2011)
Röhrig, H.: Tree decomposition: A feasibility study. Master’s thesis, Max-Planck-Institut für Informatik, Saarbrücken, Germany (1998)
Song, Y., Liu, C., Malmberg, R., Pan, F., Cai, L.: Tree decomposition based fast search of RNA structures including pseudoknots in genomes. In: Proceedings of the 2005 IEEE Computational Systems Bioinformatics Conference, CSB 2005, pp. 223–234 (2005)
Treewidthlib (2004), http://www.cs.uu.nl/people/hansb/treewidthlib
van Rooij, J.M.M., Bodlaender, H.L., Rossmanith, P.: Dynamic Programming on Tree Decompositions using Generalised Fast Subset Convolution. In: Fiat, A., Sanders, P. (eds.) ESA 2009. LNCS, vol. 5757, pp. 566–577. Springer, Heidelberg (2009)
Zhao, J., Che, D., Cai, L.: Comparative pathway annotation with protein-DNA interaction and operon information via graph tree decomposition. In: Proceedings of Pacific Symposium on Biocomputing, PSB 2007, vol. 12, pp. 496–507 (2007)
Zhao, J., Malmberg, R.L., Cai, L.: Rapid ab initio prediction of RNA pseudoknots via graph tree decomposition. Journal of Mathematical Biology 56(1-2), 145–159 (2008)
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Hvidevold, E.M., Sharmin, S., Telle, J.A., Vatshelle, M. (2012). Finding Good Decompositions for Dynamic Programming on Dense Graphs. In: Marx, D., Rossmanith, P. (eds) Parameterized and Exact Computation. IPEC 2011. Lecture Notes in Computer Science, vol 7112. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-28050-4_18
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