The classic technique of Baker [J. ACM '94] is the most fundamental approach for designing approximation schemes on planar, or more generally topologically-constrained graphs, and it has been applied in a myriad of different variants and settings throughout the last 30 years. In this work we propose a dynamic variant of Baker's technique, where instead of finding an approximate solution in a given static graph, the task is to design a data structure for maintaining an approximate solution in a fully dynamic graph, that is, a graph that is changing over time by edge deletions and edge insertions. Specifically, we address the two most basic problems -- Maximum Weight Independent Set and Minimum Weight Dominating Set -- and we prove the following: for a fully dynamic $n$-vertex planar graph $G$, one can: * maintain a $(1-\varepsilon)$-approximation of the maximum weight of an independent set in $G$ with amortized update time $f(\varepsilon)\cdot n^{o(1)}$; and, * under the additional assumption that the maximum degree of the graph is bounded at all times by a constant, also maintain a $(1+\varepsilon)$-approximation of the minimum weight of a dominating set in $G$ with amortized update time $f(\varepsilon)\cdot n^{o(1)}$. In both cases, $f(\varepsilon)$ is doubly-exponential in $\mathrm{poly}(1/\varepsilon)$ and the data structure can be initialized in time $f(\varepsilon)\cdot n^{1+o(1)}$. All our results in fact hold in the larger generality of any graph class that excludes a fixed apex-graph as a minor.

翻译：Baker [J. ACM '94] 的经典技术是设计平面图或更一般拓扑约束图近似方案的最基本方法，在过去30年中已应用于无数变体与场景。本文提出Baker技术的动态变体：任务不再是给定静态图中寻找近似解，而是设计数据结构维护全动态图（即通过边删除和边插入随时间变化的图）中的近似解。具体而言，我们针对两个最基本问题——最大权独立集与最小权支配集——证明以下结论：对于全动态$n$顶点平面图$G$，可以*以摊还更新时间$f(\varepsilon)\cdot n^{o(1)}$维护$G$中独立集最大权的$(1-\varepsilon)$-近似；*在额外假设图的最大度始终有常数上界时，以摊还更新时间$f(\varepsilon)\cdot n^{o(1)}$维护$G$中支配集最小权的$(1+\varepsilon)$-近似。两种情形中$f(\varepsilon)$关于$\mathrm{poly}(1/\varepsilon)$呈双指数增长，且数据结构可在时间$f(\varepsilon)\cdot n^{1+o(1)}$内初始化。事实上，我们的所有结果适用于排除固定顶点图作为子式的更广图类。