On the boolean-width of a graph: structure and applications
Résumé
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 $O^*(2^{O(\log ^4 n)})$.