On Feedback Vertex Set: New Measure and New Structures

TL;DR

This paper introduces a new parameterized algorithm for the feedback vertex set problem, utilizing a novel disjoint variant, improved kernelization, and a refined branch-and-search method to achieve faster solutions.

Contribution

It presents a new approach to feedback vertex set by developing an improved kernelization and a novel branch-and-search process with a new measure, leading to a faster algorithm.

Findings

Achieved an $O^*(3.83^k)$-time algorithm for FVS.

Developed an improved kernelization for disjoint-FVS.

Created a new branch-and-search measure that enhances efficiency.

Abstract

We present a new parameterized algorithm for the {feedback vertex set} problem ({\sc fvs}) on undirected graphs. We approach the problem by considering a variation of it, the {disjoint feedback vertex set} problem ({\sc disjoint-fvs}), which finds a feedback vertex set of size $k$ that has no overlap with a given feedback vertex set $F$ of the graph $G$. We develop an improved kernelization algorithm for {\sc disjoint-fvs} and show that {\sc disjoint-fvs} can be solved in polynomial time when all vertices in $G \setminus F$ have degrees upper bounded by three. We then propose a new branch-and-search process on {\sc disjoint-fvs}, and introduce a new branch-and-search measure. The process effectively reduces a given graph to a graph on which {\sc disjoint-fvs} becomes polynomial-time solvable, and the new measure more accurately evaluates the efficiency of the process. These algorithmic and combinatorial studies enable us to develop an $O^*(3.83^k)$-time parameterized algorithm for the general {\sc fvs} problem, improving all previous algorithms for the problem.