Dynamic Parameterized Problems

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We study the parameterized complexity of various 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. 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 classical $$\Pi $$ -Deletion problem and show 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 algorithms and linear kernels. Specifically, we show that Dynamic Vertex Cover has a deterministic algorithm with $$1.0822^k n^{\mathcal {O}(1)}$$ running time and Dynamic Feedback Vertex Set has a randomized algorithm with $$1.6667^k n^{\mathcal {O}(1)}$$ running time. We also show that Dynamic Connected Feedback Vertex Set can be solved in $$2^{\mathcal {O}(k)} n^{\mathcal {O}(1)}$$ time. For each of Dynamic Connected Vertex Cover, Dynamic Dominating Set and Dynamic Connected Dominating Set, we describe an algorithm with $$2^k n^{\mathcal {O}(1)}$$ running time and show that this is the optimal running time (up to polynomial factors) assuming the Set Cover Conjecture.