Abstract
The two weighted graph problems Node Multiway Cut (NMC) and Subset Feedback Vertex Set (SFVS) both ask for a vertex set of minimum total weight, that for NMC disconnects a given set of terminals, and for SFVS intersects all cycles containing a vertex of a given set. We design a meta-algorithm that allows to solve both problems in time 2^{O(rw^3)}cdot n^{4}, 2^{O(q^2log (q))}cdot n^{4}, and n^{O(k^2)} where rw is the rank-width, q the {mathbb {Q}}-rank-width, and k the mim-width of a given decomposition. This answers in the affirmative an open question raised by Jaffke et al. (Algorithmica 82(1):118–145, 2020) concerning an XP algorithm for SFVS parameterized by mim-width. By a unified algorithm, this solves both problems in polynomial-time on the following graph classes: Interval, Permutation, and Bi-Interval graphs, Circular Arc and Circular Permutation graphs, Convex graphs, k-Polygon, Dilworth-k and Co-k-Degenerate graphs for fixed k; and also on Leaf Power graphs if a leaf root is given as input, on H-Graphs for fixed H if an H-representation is given as input, and on arbitrary powers of graphs in all the above classes. Prior to our results, only SFVS was known to be tractable restricted only on Interval and Permutation graphs, whereas all other results are new.
Highlights
Given a vertex-weighted graph G and a set S of its vertices, the Subset Feedback Vertex Set (SFVS) problem asks for a vertex set of minimum weight that intersects all cycles containing a vertex of S
We resolve in the affirmative the question raised by Jaffke et al [29], mentioned in [37, 38], asking whether there is an XP-time algorithm for SFVS parameterized by the mim-width of a given decomposition
If a branch-decomposition of mim-width k for G is given as input, we can solve Subset Feedback Vertex Set and Node Multiway Cut in time nO(k2)
Summary
Given a vertex-weighted graph G and a set S of its vertices, the Subset Feedback Vertex Set (SFVS) problem asks for a vertex set of minimum weight that intersects all cycles containing a vertex of S. If a branch-decomposition of mim-width k for G is given as input, we can solve Subset Feedback Vertex Set and Node Multiway Cut in time nO(k2) Note it is not known whether the mim-width of a graph can be approximated within a constant factor in time n f (k) for some function f. Our Approach We give some intuition to our meta-algorithm, that will focus on Subset Feedback Vertex Set. Since NMC can be solved by adding a vertex v of large weight adjacent to all terminals and solving SFVS with S = {v}, all within the same runtime as extending the given branch-decomposition to this new graph increases the width at most by one for all considered width measures. We know that it is not the case for tree-width as FVS can be solved in 2O(k) · n [5] but SFVS cannot be solved in ko(k) · nO(1) unless ETH fails [2]
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