The paper deals with the Feedback Vertex Set problem parameterized by the solution size. Given a graph $G$ and a parameter $k$, one has to decide if there is a set $S$ of at most $k$ vertices such that $G-S$ is acyclic. Assuming the Exponential Time Hypothesis, it is known that FVS cannot be solved in time $2^{o(k)}n^{\mathcal{O}(1)}$ in general graphs. To overcome this, many recent results considered FVS restricted to particular intersection graph classes and provided such $2^{o(k)}n^{\mathcal{O}(1)}$ algorithms. In this paper we provide generic conditions on a graph class for the existence of an algorithm solving FVS in subexponential FPT time, i.e. time $2^{k^\varepsilon} \mathop{\rm poly}(n)$, for some $\varepsilon<1$, where $n$ denotes the number of vertices of the instance and $k$ the parameter. On the one hand this result unifies algorithms that have been proposed over the years for several graph classes such as planar graphs, map graphs, unit-disk graphs, pseudo-disk graphs, and string graphs of bounded edge-degree. On the other hand it extends the tractability horizon of FVS to new classes that are not amenable to previously used techniques, in particular intersection graphs of ``thin'' objects like segment graphs or more generally $s$-string graphs.

翻译：本文研究以解大小为参数的反馈顶点集问题。给定图$G$与参数$k$，需要判断是否存在至多$k$个顶点的集合$S$使得$G-S$是无环图。在指数时间假设下，已知对于一般图类，反馈顶点集问题无法在$2^{o(k)}n^{\mathcal{O}(1)}$时间内求解。为突破此限制，近年许多研究聚焦于特定交图类中的反馈顶点集问题，并给出了相应的$2^{o(k)}n^{\mathcal{O}(1)}$算法。本文提出图类满足亚指数FPT时间（即对某个$\varepsilon<1$，在$2^{k^\varepsilon} \mathop{\rm poly}(n)$时间内）可解反馈顶点集问题的通用条件，其中$n$表示实例顶点数，$k$为参数。该成果一方面统一了多年来针对平面图、地图图、单位圆盘图、伪圆盘图及有界边度字符串图等多个图类提出的算法；另一方面将反馈顶点集的可解范围拓展至先前技术无法处理的新图类，特别是"薄型"对象（如线段图或更一般的$s$-字符串图）的交图类。