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

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 $\mathrm{NP}$-hard even in split and chordal graphs, and SFVS in Chordal Graphs can be considered as a special case of the 3-Hitting Set problem. However, it is not easy to solve SFVS in Chordal Graphs faster than 3-Hitting Set. In 2019, Philip, Rajan, Saurabh, and Tale (Algorithmica 2019) proved that SFVS in Chordal Graphs can be solved in $2^k n^{\mathcal{O}(1)}$, slightly improving the best result $2.076^k n^{\mathcal{O}(1)}$ for 3-Hitting Set. In this paper, we break the "$2^k$-barrier" for SFVS in Chordal Graphs by giving a $1.619^k n^{\mathcal{O}(1)}$-time algorithm. Our algorithm uses reduction and branching rules based on the Dulmage-Mendelsohn decomposition and a divide-and-conquer method.Comment: 27 pages, 8 figures. Full versio