Mim-Width II. The Feedback Vertex Set Problem

Abstract

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 nO(w)-time algorithm that solves FEEDBACK VERTEX SET. This provides a unified polynomial-time algorithm for many well-known classes, such as INTERVAL graphs, PERMUTATION graphs, and LEAF POWER graphs (given a leaf root), and furthermore, it gives the first polynomial-time algorithms for other classes of bounded mim-width, such as CIRCULAR PERMUTATION and CIRCULAR k-TRAPEZOID graphs (given a circular k-trapezoid model) for fixed k. We complement our result by showing that FEEDBACK VERTEX SET is W[1]-hard when parameterized by w and the hardness holds even when a linear branch decomposition of mim-width w is given.