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 (SFVS-C) can be considered as a special case of the 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 $2^k n^{\mathcal{O}(1)}$, 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-C 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.

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，使得剩余图中任意顶点子集（称为终端集）中的顶点均不位于环上，该问题推广了著名的反馈顶点集问题和多路割问题。SFVS在分裂图和弦图中仍为 $\mathrm{NP}$-困难问题，而弦图中的SFVS（SFVS-C）可视为3-击中集问题的特例。然而，要使SFVS-C的求解速度快于3-击中集问题并非易事。2019年，Philip、Rajan、Saurabh和Tale（Algorithmica 2019）证明了SFVS-C可在 $2^k n^{\mathcal{O}(1)}$ 时间内求解，小幅改进了3-击中集问题的最优结果 $2.076^k n^{\mathcal{O}(1)}$。本文通过提出一个 $1.619^k n^{\mathcal{O}(1)}$ 时间算法，打破了SFVS-C的"$2^k$-障碍"。我们的算法采用了基于杜拉奇-门德尔松分解的归约与分支规则，并融合了分治方法。