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，使得顶点子集（称为终端集）中的任意顶点在剩余图中不处于任何环内，该问题推广了著名的反馈顶点集问题与多路割问题。即使在分裂图与弦图中，SFVS仍是NP难问题，而弦图中的子集反馈顶点集问题（SFVS-C）可视为隐式的3-命中集问题。然而，要获得比3-命中集更快的SFVS-C求解算法并非易事。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，这较3-命中集的最佳结果$\mathcal{O}^{*}(2.076^{k})$略有改进。本文通过提出$\mathcal{O}^{*}(1.820^{k})$时间算法，突破了SFVS-C的"$2^{k}$障碍"。我们的算法基于Dulmage-Mendelsohn分解的规约与分支规则，并采用了分治策略。