Breaking the Barrier $2^k$ for Subset Feedback Vertex Set in Chordal Graphs

TL;DR

This paper presents a faster algorithm for the Subset Feedback Vertex Set problem in chordal graphs, improving the exponential time complexity from 2^k to approximately 1.82^k, using advanced decomposition and divide-and-conquer techniques.

Contribution

The authors develop an $ ilde{O}(1.82^k)$-time algorithm for SFVS in chordal graphs, breaking the previous 2^k barrier and employing Dulmage-Mendelsohn decomposition and divide-and-conquer strategies.

Findings

Achieved an $ ilde{O}(1.82^k)$ algorithm for SFVS-C.

Improved the exponential time complexity over previous results.

Utilized Dulmage-Mendelsohn decomposition in algorithm design.

Abstract

The Subset Feedback Vertex Set problem (SFVS), to delete $k$ vertices from a given graph such that any vertex in a vertex subset (called a terminal set) is not in a cycle in the remaining graph, generalizes the famous Feedback Vertex Set problem and Multiway Cut problem. SFVS remains NP-hard even in split and chordal graphs, and SFVS in Chordal Graphs (SFVS-C) can be considered as an implicit 3-Hitting Set problem. However, it is not easy to solve SFVS-C faster than 3-Hitting Set. In 2019, Philip, Rajan, Saurabh, and Tale (Algorithmica 2019) proved that SFVS-C can be solved in $\mathcal{O}^{*}(2^{k})$ time, slightly improving the best result $\mathcal{O}^{*}(2.076^{k})$ for 3-Hitting Set. In this paper, we break the "$2^{k}$-barrier" for SFVS-C by giving an $\mathcal{O}^{*}(1.820^{k})$-time algorithm. Our algorithm uses reduction and branching rules based on the Dulmage-Mendelsohn decomposition and a divide-and-conquer method.