Dynamic Meta-Kernelization

Abstract

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.