Directed Feedback Vertex Set is Fixed-Parameter Tractable

TL;DR

This paper proves that the Directed Feedback Vertex Set problem is fixed-parameter tractable by presenting an algorithm with runtime exponential in the parameter but polynomial in the input size.

Contribution

It provides the first fixed-parameter algorithm for the Directed Feedback Vertex Set problem, resolving a long-standing open question.

Findings

Algorithm runs in O(8^k k! poly(n)) time.

Confirms fixed-parameter tractability of the problem.

Advances understanding of directed graph feedback vertex sets.

Abstract

We resolve positively a long standing open question regarding the fixed-parameter tractability of the parameterized Directed Feedback Vertex Set problem. In particular, we propose an algorithm which solves this problem in $O(8^kk!*poly(n))$.