A Naive Algorithm for Feedback Vertex Set

TL;DR

This paper presents a simple greedy branching algorithm for the feedback vertex set problem that operates in single-exponential time, improving understanding of its computational complexity.

Contribution

It introduces a naive yet effective greedy branching algorithm with proven single-exponential time complexity for the feedback vertex set problem.

Findings

Algorithm runs in $O(c^k imes n^2)$ time

Branching on highest degree vertices is effective

Provides insights into fixed-parameter tractability

Abstract

Given a graph on $n$ vertices and an integer $k$, the feedback vertex set problem asks for the deletion of at most $k$ vertices to make the graph acyclic. We show that a greedy branching algorithm, which always branches on an undecided vertex with the largest degree, runs in single-exponential time, i.e., $O(c^k\cdot n^2)$ for some constant $c$.