On the boolean-width of a graph: structure and applications

Isolde Adler 1 Binh-Minh Bui-Xuan 2, 1 Yuri Rabinovich 3 Gabriel Renault 1 Jan Arne Telle 1 Martin Vatshelle 1
1 Department of Informatics
2 APR - Algorithmes, Programmes et Résolution
LIP6 - Laboratoire d'Informatique de Paris 6
3 Department of Computer Science [Haifa]
Abstract : Boolean-width is a recently introduced graph invariant. Similar to tree-width, it measures the structural complexity of graphs. Given any graph $G$ and a decomposition of $G$ of boolean-width $k$, we give algorithms solving a large class of vertex subset and vertex partitioning problems in time $O^*(2^{O(k^2)})$. We relate the boolean-width of a graph to its branch-width and to the boolean-width of its incidence graph. For this we use a constructive proof method that also allows much simpler proofs of similar results on rank-width in [S. Oum. Rank-width is less than or equal to branch-width. \emph{Journal of Graph Theory} 57(3):239--244, 2008]. For an $n$-vertex random graph, with a uniform edge distribution, we show that almost surely its boolean-width is $\Theta(\log^2 n)$ -- setting boolean-width apart from other graph invariants -- and it is easy to find a decomposition witnessing this. Combining our results gives algorithms that on input a random graph on $n$ vertices will solve a large class of vertex subset and vertex partitioning problems in quasi-polynomial time $O^*(2^{O(\log ^4 n)})$.
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Conference papers
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https://hal.archives-ouvertes.fr/hal-00555502
Contributor : Binh-Minh Bui-Xuan <>
Submitted on : Thursday, January 13, 2011 - 3:43:52 PM
Last modification on : Thursday, March 21, 2019 - 1:05:48 PM

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Isolde Adler, Binh-Minh Bui-Xuan, Yuri Rabinovich, Gabriel Renault, Jan Arne Telle, et al.. On the boolean-width of a graph: structure and applications. 36th International Workshop on Graph-Theoretic Concepts in Computer Science (WG'10), Jun 2010, Zarós, Crete, Greece. pp.159-170, ⟨10.1007/978-3-642-16926-7_16⟩. ⟨hal-00555502⟩

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