结构舍入：图在数学上可处理类附近的近似算法
Structural Rounding: Approximation Algorithms for Graphs Near an Algorithmically Tractable Class
摘要：
我们开发了一个新的框架，用于从结构图算法文献中概括近似算法，以便它们适用于与该类有点接近的图（我们期望在使用真实网络时常见的情况），同时仍然保证近似比。其思想是通过顶点删除或边删除对给定图进行$\textit{edit}$，将图放入算法上易于处理的类中，对该类应用已知的近似算法，然后将$\textit{lift}$的解决方案应用于原始图。当一个优化问题是服从这种方法，我们给出了一个一般的表征，并表明它包括许多研究良好的图形问题，如独立集，顶点覆盖，反馈顶点集，最小最大匹配，色数，（$\ell$-）支配集，边（$\ell$ -）支配集，连通支配集。为了实现这个框架，我们开发了新的编辑算法，找到了将给定的图形带入几个重要图形类之一所需的近似最少的编辑（在某些情况下，也近似于家庭的目标参数）。对于有界退化，我们得到一个双准则$(4,4)$ -近似，也扩展到一个更平滑的双准则权衡。对于有界树宽，我们得到了一个双准则$(O(\log^{1.5} n), O(\sqrt{\log w}))$ -近似，对于有界路宽，我们得到了一个双准则$(O(\log^{1.5} n), O(\sqrt{\log w} \cdot \log n))$ -近似.对于树深度$2$（也与有界扩展有关），我们得到了一个$4$ -近似。我们还证明了互补的硬度近似结果假设$\mathrm{P} \neq \mathrm{NP}$：特别是，这些问题都是对数因子不可近似，除了最后一个是不可近似低于一些常数因子（$2$假设UGC）。
英文摘要：
We develop a new framework for generalizing approximation algorithms from the structural graph algorithm literature so that they apply to graphs somewhat close to that class (a scenario we expect is common when working with real-world networks) while still guaranteeing approximation ratios. The idea is to $\textit{edit}$ a given graph via vertex- or edge-deletions to put the graph into an algorithmically tractable class, apply known approximation algorithms for that class, and then $\textit{lift}$ the solution to apply to the original graph. We give a general characterization of when an optimization problem is amenable to this approach, and show that it includes many well-studied graph problems, such as Independent Set, Vertex Cover, Feedback Vertex Set, Minimum Maximal Matching, Chromatic Number, ($\ell$-)Dominating Set, Edge ($\ell$-)Dominating Set, and Connected Dominating Set. To enable this framework, we develop new editing algorithms that find the approximately-fewest edits required to bring a given graph into one of several important graph classes (in some cases, also approximating the target parameter of the family). For bounded degeneracy, we obtain a bicriteria $(4,4)$-approximation which also extends to a smoother bicriteria trade-off. For bounded treewidth, we obtain a bicriteria $(O(\log^{1.5} n), O(\sqrt{\log w}))$-approximation, and for bounded pathwidth, we obtain a bicriteria $(O(\log^{1.5} n), O(\sqrt{\log w} \cdot \log n))$-approximation. For treedepth $2$ (also related to bounded expansion), we obtain a $4$-approximation. We also prove complementary hardness-of-approximation results assuming $\mathrm{P} \neq \mathrm{NP}$: in particular, these problems are all log-factor inapproximable, except the last which is not approximable below some constant factor ($2$ assuming UGC).
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