Abstract
We introduce the graph parameter boolean-width, related to the number of different unions of neighborhoods across a cut of a graph. For many graph problems this number is the runtime bottleneck when using a divide-and-conquer approach. Boolean-width is similar to rank-width, which is related to the number of GF(2)-sums (1+1=0) of neighborhoods instead of the Boolean-sums (1+1=1) used for boolean-width. For an n-vertex graph G given with a decomposition tree of boolean-width k we show how to solve Minimum Dominating Set, Maximum Independent Set and Minimum or Maximum Independent Dominating Set in time O(n(n + 23k k )). We show for any graph that its boolean-width is never more than the square of its rank-width. We also exhibit a class of graphs, the Hsu-grids, having the property that a Hsu-grid on Θ(n 2) vertices has boolean-width Θ(logn) and tree-width, branch-width, clique-width and rank-width Θ(n). Moreover, any optimal rank-decomposition of such a graph will have boolean-width Θ(n), i.e. exponential in the optimal boolean-width.
Supported by the Norwegian Research Council, project PARALGO.
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References
Adler, I., Vatshelle, M.: Personal communication
Bodlaender, H., Koster, A.: Treewidth Computations I Upper Bounds. Technical Report UU-CS-2008-032, Department of Information and Computing Sciences, Utrecht University (2008)
Brandstaedt, A., Lozin, V.V.: On the linear structure and clique-width of bipartite permutation graphs. Ars Combinatoria 67, 719–734 (2003)
Bui-Xuan, B.-M., Telle, J.A., Vatshelle, M.: Fast FPT algorithms for vertex subset and vertex partitioning problems using neighborhood unions, http://arxiv.org/abs/0903.4796+
Bui-Xuan, B.-M., Telle, J.A., Vatshelle, M.: H-join decomposable graphs and algorithms with runtime single exponential in rankwidth. To appear in DAM: special issue of GROW, http://www.ii.uib.no/~telle/bib/BTV.pdf
Corneil, D., Rotics, U.: On the relationship between clique-width and treewidth. SIAM Journal on Computing 34(4), 825–847 (2005)
Damm, C., Kim, K.H., Roush, F.W.: On covering and rank problems for boolean matrices and their applications. In: Asano, T., Imai, H., Lee, D.T., Nakano, S.-i., Tokuyama, T. (eds.) COCOON 1999. LNCS, vol. 1627, pp. 123–133. Springer, Heidelberg (1999)
Dorn, F.: Dynamic programming and fast matrix multiplication. In: Azar, Y., Erlebach, T. (eds.) ESA 2006. LNCS, vol. 4168, pp. 280–291. Springer, Heidelberg (2006)
Ganian, R., Hliněný, P.: On Parse Trees and Myhill-Nerode-type Tools for handling Graphs of Bounded Rank-width (submitted manuscript), http://www.fi.muni.cz/~hlineny/Research/papers/MNtools+dam3.pdf
Geelen, J., Gerards, A., Whittle, G.: Branch-width and well-quasi-ordering in matroids and graphs. Journal of Combinatorial Theory, Series B 84(2), 270–290 (2002)
Hliněný, P., Oum, S.: Finding branch-decompositions and rank-decompositions. SIAM Journal on Computing 38(3), 1012–1032 (2008); Abstract at ESA 2007.
Hliněný, P., Oum, S., Seese, D., Gottlob, G.: Width parameters beyond tree-width and their applications. The Computer Journal 51(3), 326–362 (2008)
Hsu, W.-L.: Decomposition of perfect graphs. Journal of Combinatorial Theory, Series B 43(1), 70–94 (1987)
Jelínek, V.: The rank-width of the square grid. In: Broersma, H., Erlebach, T., Friedetzky, T., Paulusma, D. (eds.) WG 2008. LNCS, vol. 5344, pp. 230–239. Springer, Heidelberg (2008)
Kim, K.H.: Boolean matrix theory and its applications. Marcel Dekker, New York (1982)
Kobler, D., Rotics, U.: Edge dominating set and colorings on graphs with fixed clique-width. Discrete Applied Mathematics 126(2-3), 197–221 (2003); Abstract at SODA 2001
Nguyen, H.X., Thiran, P.: Active measurement for multiple link failures diagnosis in IP networks. In: Barakat, C., Pratt, I. (eds.) PAM 2004. LNCS, vol. 3015, pp. 185–194. Springer, Heidelberg (2004)
Oum, S.: Graphs of Bounded Rank-width. PhD thesis, Princeton University (2005)
Oum, S.: Rank-width is less than or equal to branch-width. Journal of Graph Theory 57(3), 239–244 (2008)
Oum, S., Seymour, P.: Approximating clique-width and branch-width. Journal of Combinatorial Theory, Series B 96(4), 514–528 (2006)
Pattison, P., Breiger, R.: Lattices and dimensional representations: matrix decompositions and ordering structures. Social Networks 24(4), 423–444 (2002)
Robertson, N., Seymour, P.: Graph minors. X. Obstructions to tree-decomposition. Journal of Combinatorial Theory, Series B 52(2), 153–190 (1991)
Rooij, J., Bodlaender, H., 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)
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Bui-Xuan, B.M., Telle, J.A., Vatshelle, M. (2009). Boolean-Width of Graphs. In: Chen, J., Fomin, F.V. (eds) Parameterized and Exact Computation. IWPEC 2009. Lecture Notes in Computer Science, vol 5917. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-11269-0_5
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