动态元核化
Dynamic Meta-Kernelization
AI总结：
本文将稀疏图类上经典NP-hard问题的线性核化结果推广至动态环境，提出动态数据结构在拓扑无小图中维护近似最优凸出分解，并由此导出动态核化、常数近似及改进的动态FPT算法。
摘要：
核化研究多项式时间预处理算法。在过去20年中，该领域最著名的正面成果是针对稀疏图类上经典NP-hard图问题的线性核。在本文中，我们将这些结果提升到动态环境中。作为典型示例，Alber、Fellows和Niedermeier [J. ACM 2004] 给出了平面图上支配集的线性核。我们提供了其核的如下动态版本：我们的数据结构以 $O(n \log n)$ 摊还时间初始化一个 $n$ 顶点平面图 $G$，并在初始化时输出一个平面图 $K$，满足 $\mathrm{OPT}(K) = \mathrm{OPT}(G)$ 且 $|K| = O(\mathrm{OPT}(G))$，其中 $\mathrm{OPT}(\cdot)$ 表示最小支配集的大小。在 $G$ 保持平面性的承诺下，可通过插入和删除边及孤立顶点以每次更新 $O(\log n)$ 摊还时间更新图 $G$。每次更新 $G$ 后，数据结构输出 $O(1)$ 次对 $K$ 的更新，维持 $\mathrm{OPT}(K) = \mathrm{OPT}(G)$、$|K| = O(\mathrm{OPT}(G))$ 以及 $K$ 的平面性。此外，对于在（拓扑）无小图类上满足特定条件的所有问题，我们获得了类似的动态核化算法。除核化外，这还直接为这些问题蕴含了新的动态常数近似算法以及对动态FPT算法的改进。我们的主要技术贡献是设计了一种动态数据结构，用于维护动态拓扑无小图的近似最优凸出分解。凸出分解由Bodlaender、Fomin、Lokshtanov、Penninkx、Saurabh和Thilikos [J. ACM 2016] 引入，此后已发展成为核化与参数化算法核心工具箱的一部分。
英文摘要：
Kernelization studies polynomial-time preprocessing algorithms. Over the last 20 years, the most celebrated positive results of the field have been linear kernels for classical NP-hard graph problems on sparse graph classes. In this paper, we lift these results to the dynamic setting. As the canonical example, Alber, Fellows, and Niedermeier [J. ACM 2004] gave a linear kernel for dominating set on planar graphs. We provide the following dynamic version of their kernel: Our data structure is initialized with an $n$-vertex planar graph $G$ in $O(n \log n)$ amortized time, and, at initialization, outputs a planar graph $K$ with $\mathrm{OPT}(K) = \mathrm{OPT}(G)$ and $|K| = O(\mathrm{OPT}(G))$, where $\mathrm{OPT}(\cdot)$ denotes the size of a minimum dominating set. The graph $G$ can be updated by insertions and deletions of edges and isolated vertices in $O(\log n)$ amortized time per update, under the promise that it remains planar. After each update to $G$, the data structure outputs $O(1)$ updates to $K$, maintaining $\mathrm{OPT}(K) = \mathrm{OPT}(G)$, $|K| = O(\mathrm{OPT}(G))$, and planarity of $K$. Furthermore, we obtain similar dynamic kernelization algorithms for all problems satisfying certain conditions on (topological-)minor-free graph classes. Besides kernelization, this directly implies new dynamic constant-approximation algorithms and improvements to dynamic FPT algorithms for such problems. Our main technical contribution is a dynamic data structure for maintaining an approximately optimal protrusion decomposition of a dynamic topological-minor-free graph. Protrusion decompositions were introduced by Bodlaender, Fomin, Lokshtanov, Penninkx, Saurabh, and Thilikos [J. ACM 2016], and have since developed into a part of the core toolbox in kernelization and parameterized algorithms.
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