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.

翻译：子集反馈顶点集问题（SFVS）要求从给定图中删除$k$个顶点，使得任意属于顶点子集（称为终端集）的顶点都不在剩余图中的环上，该问题推广了著名的反馈顶点集问题和多路割问题。SFVS即使在分裂图和弦图中仍保持NP困难性，而弦图中的SFVS（SFVS-C）可视为隐式3-击中集问题。然而，以快于3-击中集的速度求解SFVS-C并非易事。2019年，Philip、Rajan、Saurabh与Tale（Algorithmica 2019）证明了SFVS-C可在$\mathcal{O}^{*}(2^{k})$时间内求解，略微改进了3-击中集问题的最佳结果$\mathcal{O}^{*}(2.076^{k})$。本文通过给出$\mathcal{O}^{*}(1.820^{k})$时间算法，突破了SFVS-C的"$2^{k}$屏障"。该算法基于Dulmage-Mendelsohn分解的归约与分支规则，并采用分治方法。