On Directed Feedback Vertex Set parameterized by treewidth

TL;DR

This paper investigates the complexity of the Directed Feedback Vertex Set problem parameterized by treewidth, establishing tight bounds on its computational complexity and providing algorithms for both general and planar graphs.

Contribution

It proves tight lower bounds under ETH and presents matching dynamic programming algorithms, with improved results for planar graphs.

Findings

No sub-exponential algorithm for general graphs unless ETH fails.

Matching upper bounds with dynamic programming algorithms.

Improved algorithms for planar graphs with exponential dependence on treewidth.

Abstract

We study the Directed Feedback Vertex Set problem parameterized by the treewidth of the input graph. We prove that unless the Exponential Time Hypothesis fails, the problem cannot be solved in time $2^{o(t\log t)}\cdot n^{\mathcal{O}(1)}$ on general directed graphs, where $t$ is the treewidth of the underlying undirected graph. This is matched by a dynamic programming algorithm with running time $2^{\mathcal{O}(t\log t)}\cdot n^{\mathcal{O}(1)}$. On the other hand, we show that if the input digraph is planar, then the running time can be improved to $2^{\mathcal{O}(t)}\cdot n^{\mathcal{O}(1)}$.