# Dynamic Parameterized Problems

Abhishek Sahu, 13 Dec 2021

In this work, we study the parameterized complexity of various classical graph-theoretic problems in the dynamic framework where the input graph is being updated by a sequence of edge additions and deletions. Vertex subset problems on graphs typically deal with finding a subset of vertices having certain properties that are of interest to us. In real-world applications, the graph under consideration often changes over time and due to this dynamics, the solution at hand might lose the desired properties. The goal in the area of dynamic graph algorithms is to efficiently maintain a solution under these changes. Recomputing a new solution on the new graph is an expensive task especially when the number of modifications made to the graph is significantly smaller than the size of the graph. In the context of parameterized algorithms, two natural parameters are the size \(k\) of the symmetric difference of the edge sets of the two graphs (on \(n\) vertices) and the size \(r\) of the symmetric difference of the two solutions. We study the Dynamic Pi-Deletion problem which is the dynamic variant of the Pi-Deletion problem and show \(\mathsf{NP}\)-hardness, fixed-parameter tractability and kernelization results. For specific cases of Dynamic Pi-Deletion such as Dynamic Vertex Cover and Dynamic Feedback Vertex Set, we describe improved \(\mathsf{FPT}\) algorithms and give linear kernels. Specifically, we show that Dynamic Vertex Cover admits algorithms with running times \(1.1740^k\cdot n^{O(1)}\) (polynomial space) and \(1.1277^k\cdot n^{O(1)}\) (exponential space). Then, we show that Dynamic Feedback Vertex Set admits a randomized algorithm with \(1.6667^k\cdot n^{O(1)}\) running time. Finally, we consider Dynamic Connected Vertex Cover, Dynamic Dominating Set and Dynamic Connected Dominating Set and describe algorithms with \(2^k\cdot n^{O(1)}\) running time improving over the known running time bounds for these problems. Additionally, for Dynamic Dominating Set and Dynamic Connected Dominating Set, we show that this is the optimal running time (up to polynomial factors) assuming the Set Cover Conjecture.

Joint work with R. Krithika, and Prafullkumar Tale