In this paper we introduce Sampling with a Black Box, a generic technique for the design of parameterized approximation algorithms for vertex deletion problems (e.g., Vertex Cover, Feedback Vertex Set, etc.). The technique relies on two components: $\bullet$ A Sampling Step. A polynomial time randomized algorithm which given a graph $G$ returns a random vertex $v$ such that the optimum of $G\setminus \{v\}$ is smaller by $1$ than the optimum of $G$ with some prescribed probability $q$. We show such algorithms exists for multiple vertex deletion problems. $\bullet$ A Black Box algorithm which is either an exact parameterized algorithm or a polynomial time approximation algorithm. Our technique combines these two components together. The sampling step is applied iteratively to remove vertices from the input graph, and then the solution is extended using the black box algorithm. The process is repeated sufficiently many times so that the target approximation ratio is attained with a constant probability. The main novelty of our work lies in the analysis of the framework and the optimization of the parameters it uses. We use the technique to derive parameterized approximation algorithm for several vertex deletion problems, including Feedback Vertex Set, $d$-Hitting Set and $\ell$-Path Vertex Cover. In particular, for every approximation ratio $1<\beta<2$, we attain a parameterized $\beta$-approximation for Feedback Vertex Set which is faster than the parameterized $\beta$-approximation of [Jana, Lokshtanov, Mandal, Rai and Saurabh, MFCS 23']. Furthermore, our algorithms are always faster than the algorithms attained using Fidelity Preserving Transformations [Fellows, Kulik, Rosamond, and Shachnai, JCSS 18'].

翻译：本文提出一种名为"黑盒采样"的通用技术，用于设计顶点删除问题（如顶点覆盖、反馈顶点集等）的参数化近似算法。该技术依赖于两个核心组件：$\bullet$ 采样步骤。这是一个多项式时间随机算法，给定图$G$，该算法以预设概率$q$返回随机顶点$v$，使得$G\setminus \{v\}$的最优解比$G$的最优解小$1$。我们证明了多个顶点删除问题均存在此类算法。$\bullet$ 黑盒算法。该算法可以是精确参数化算法或多项式时间近似算法。我们的技术将这两个组件相结合：通过迭代应用采样步骤从输入图中移除顶点，随后利用黑盒算法扩展解。此过程重复足够多次，从而以恒定概率达到目标近似比。本工作的主要创新在于对该框架的分析及其参数的优化。我们运用该技术为多个顶点删除问题推导出参数化近似算法，包括反馈顶点集、$d$-命中集和$\ell$-路径顶点覆盖。特别地，对于任意近似比$1<\beta<2$，我们获得了比[Jana, Lokshtanov, Mandal, Rai and Saurabh, MFCS 23']更快的反馈顶点集参数化$\beta$-近似算法。此外，我们的算法始终比使用保真度保持变换[Fellows, Kulik, Rosamond, and Shachnai, JCSS 18']获得的算法更高效。