A note on the complexity of Feedback Vertex Set parameterized by mim-width

TL;DR

This paper demonstrates that the Feedback Vertex Set problem remains W[1]-hard when parameterized by mim-width, even in restricted graph classes, contrasting with recent XP-time algorithms.

Contribution

It establishes the W[1]-hardness of Feedback Vertex Set parameterized by mim-width, extending the understanding of its computational complexity.

Findings

Feedback Vertex Set is W[1]-hard for linear mim-width.

Hardness persists for H-graphs with edge-based parameters.

Contrasts with recent XP-time algorithms for the same parameter.

Abstract

We complement the recent algorithmic result that Feedback Vertex Set is XP-time solvable parameterized by the mim-width of a given branch decomposition of the input graph [3] by showing that the problem is W[1]-hard in this parameterization. The hardness holds even for linear mim-width, as well as for H-graphs, where the parameter is the number of edges in H. To obtain this result, we adapt a reduction due to Fomin, Golovach and Raymond [2], following the same line of reasoning but adding a new gadget.