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

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.