For numerous graph problems in the realm of parameterized algorithms, using the size of a smallest deletion set (called a modulator) into well-understood graph families as parameterization has led to a long and successful line of research. Recently, however, there has been an extensive study of structural parameters that are potentially much smaller than the modulator size. In particular, recent papers [Jansen et al. STOC 2021; Agrawal et al. SODA 2022] have studied parameterization by the size of the modulator to a graph family $\mathcal{H}$ ($\textbf{mod}_{\mathcal{H}}$), elimination distance to $\mathcal{H}$ ($\textbf{ed}_{\mathcal{H}}$), and $\mathcal{H}$-treewidth ($\textbf{tw}_{\mathcal{H}}$). While these new parameters have been successfully exploited to design fast exact algorithms their utility (especially that of latter two) in the context of approximation algorithms is mostly unexplored. The conceptual contribution of this paper is to present novel algorithmic meta-theorems that expand the impact of these structural parameters to the area of FPT Approximation, mirroring their utility in the design of exact FPT algorithms. Precisely, we show that if a covering or packing problem is definable in Monadic Second Order Logic and has a property called Finite Integer Index, then the existence of an FPT Approximation Scheme (FPT-AS, i.e., ($1\pm \epsilon$)-approximation) parameterized these three parameters is in fact equivalent. As concrete exemplifications of our meta-theorems, we obtain FPT-ASes for well-studied graph problems such as Vertex Cover, Feedback Vertex Set, Cycle Packing and Dominating Set, parameterized by these three parameters.

翻译：针对参数化算法领域的众多图问题，将最小删除集（称为调制器）的大小作为参数化到已知图族的方法，已催生出一条长期且成功的思路。然而近年来，对可能远小于调制器大小的结构参数的研究日益深入。特别是近期论文 [Jansen 等人，STOC 2021；Agrawal 等人，SODA 2022] 研究了基于到图族 $\mathcal{H}$ 的调制器大小（$\textbf{mod}_{\mathcal{H}}$）、到 $\mathcal{H}$ 的消去距离（$\textbf{ed}_{\mathcal{H}}$）以及 $\mathcal{H}$-树宽（$\textbf{tw}_{\mathcal{H}}$）的参数化方法。尽管这些新参数已被成功用于设计快速精确算法，但在近似算法背景下其效用（尤其是后两者）尚未得到充分探索。本文的理论贡献在于提出新颖的算法元定理，将这些结构参数的影响拓展至FPT近似领域，从而镜像其在精确FPT算法设计中的实用性。具体而言，我们证明：若某覆盖或打包问题可在一阶逻辑（Monadic Second Order Logic）中定义且具有有限整数索引（Finite Integer Index）性质，则基于这三个参数化存在FPT近似方案（FPT-AS，即 $(1\pm \epsilon)$ 近似）实际上是等价的。作为元定理的具体实例，我们针对顶点覆盖（Vertex Cover）、反馈顶点集（Feedback Vertex Set）、环打包（Cycle Packing）和支配集（Dominating Set）等经典图问题，基于这三个参数化得到了FPT近似方案。