Fast Dynamic Graph Algorithms for Parameterized Problems

Fully dynamic graph is a data structure that (1) supports edge insertions and deletions and (2) answers problem specific queries. The time complexity of (1) and (2) are referred to as the update time and the query time respectively. There are many researches on dynamic graphs whose update time and query time are $o(|G|)$, that is, sublinear in the graph size. However, almost all such researches are for problems in P. In this paper, we investigate dynamic graphs for NP-hard problems exploiting the notion of fixed parameter tractability (FPT). We give dynamic graphs for Vertex Cover and Cluster Vertex Deletion parameterized by the solution size $k$. These dynamic graphs achieve almost the best possible update time $O(\mathrm{poly}(k)\log n)$ and the query time $O(f(\mathrm{poly}(k),k))$, where $f(n,k)$ is the time complexity of any static graph algorithm for the problems. We obtain these results by dynamically maintaining an approximate solution which can be used to construct a small problem kernel. Exploiting the dynamic graph for Cluster Vertex Deletion, as a corollary, we obtain a quasilinear-time (polynomial) kernelization algorithm for Cluster Vertex Deletion. Until now, only quadratic time kernelization algorithms are known for this problem. We also give a dynamic graph for Chromatic Number parameterized by the solution size of Cluster Vertex Deletion, and a dynamic graph for bounded-degree Feedback Vertex Set parameterized by the solution size. Assuming the parameter is a constant, each dynamic graph can be updated in $O(\log n)$ time and can compute a solution in $O(1)$ time. These results are obtained by another approach.Comment: SWAT 2014 to appea

Fully-dynamic Planarity Testing in Polylogarithmic Time

Given a dynamic graph subject to insertions and deletions of edges, a natural question is whether the graph presently admits a planar embedding. We give a deterministic fully-dynamic algorithm for general graphs, running in amortized $O(\log^3 n)$ time per edge insertion or deletion, that maintains a bit indicating whether or not the graph is presently planar. This is an exponential improvement over the previous best algorithm [Eppstein, Galil, Italiano, Spencer, 1996] which spends amortized $O(\sqrt{n})$ time per update.Comment: Updated version of paper submitted to STOC'20. This version features a complete rewrite of section 4.4 (do-separation-flips). The new version fixes an overlooked case in the previous version (the two fundamental cycles we find do not necessarily share an edge) and contains a detailed case-by-case proof of correctnes