Node Multiway Cut and Subset Feedback Vertex Set on Graphs of Bounded Mim-width

TL;DR

This paper presents a unified algorithmic framework for solving Node Multiway Cut and Subset Feedback Vertex Set problems on graphs with bounded mim-width, answering open questions and extending tractability to many graph classes.

Contribution

It introduces a meta-algorithm that solves both problems efficiently on graphs with bounded mim-width, Q-rank-width, and rank-width, and applies to numerous graph classes, expanding known polynomial-time solvability.

Findings

Polynomial-time algorithms for SFVS and NMC on various graph classes.

Answering an open question about XP algorithms for SFVS parameterized by mim-width.

Extension of tractability results to new graph classes.

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 $2^{O(rw^3)}\cdot n^{4}$, $2^{O(q^2\log(q))}\cdot n^{4}$, 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, 2019) 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.