3 years ago

# A unified polynomial-time algorithm for Feedback Vertex Set on graphs of bounded mim-width.

Jan Arne Telle, Lars Jaffke, O-joung Kwon

We give a first polynomial-time algorithm for (Weighted) Feedback Vertex Set on graphs of bounded maximum induced matching width (mim-width). Explicitly, given a branch decomposition of mim-width $w$, we give an $n^{\mathcal{O}(w)}$-time algorithm that solves Feedback Vertex Set. This provides a unified algorithm for many well-known classes, such as Interval graphs and Permutation graphs, and furthermore, it gives the first polynomial-time algorithms for other classes of bounded mim-width, such as Circular Permutation, Circular $k$-Trapezoid and Dilworth-$k$ graphs for fixed $k$. In all these classes the decomposition is computable in polynomial time, as shown by Belmonte and Vatshelle [Theor. Comput. Sci. 2013]. We show that powers of graphs of tree-width $w + 1$ and powers of graphs of clique-width $w$ have mim-width at most $w$. These results extensively provide new classes of bounded mim-width. The first result implies that, surprisingly, Leaf Power graphs which are of importance in the field of phylogenetic studies have mim-width at most $1$. Given a tree-decomposition of width $w + 1$ or a clique-width $w$-expression of a graph $G$, one can for any value of $k$ find a mim-width decomposition of its $k$-power in polynomial time, and apply our algorithm to solve Feedback Vertex Set on the k-power in time $n^{\mathcal{O}(w)}$. In contrast to Feedback Vertex Set, we show that Hamiltonian Cycle is NP-complete even on graphs of linear mim-width $1$, which further hints at the expressive power of the mim-width parameter.

Publisher URL: http://arxiv.org/abs/1710.07148

DOI: arXiv:1710.07148v1

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