On the parameterized complexity of dynamic problems

In a dynamic version of a (base) problem X it is assumed that some solution to an instance of X is no longer feasible due to changes made to the original instance, and it is required that a new feasible solution be obtained from what “remained” from the original solution at a minimal cost. In the parameterized version of such a problem, the changes made to an instance are bounded by an edit-parameter, while the cost of reconstructing a solution is bounded by some increment-parameter.Capitalizing on the recent initial work of Downey et al. on the Dynamic Dominating Set problem, we launch a study of the dynamic versions of a number of problems including Vertex Cover, Maximum Clique, Connected Vertex Cover and Connected Dominating Set. In particular, we show that Dynamic Vertex Cover is W[1]W[1]-hard, and the connected versions of both Dynamic Vertex Cover and Dynamic Dominating Set become fixed-parameter tractable with respect to the edit-parameter while they remain W[2]W[2]-hard with respect to the increment-parameter. Moreover, we show that Dynamic Independent Dominating Set is W[2]W[2]-hard with respect to the edit-parameter.We introduce the reoptimization parameter, which bounds the difference between the cardinalities of initial and target solutions. We prove that, while Dynamic Maximum Clique is fixed-parameter tractable with respect to the edit-parameter, it becomes W[1]W[1]-hard if the increment-parameter is replaced with the reoptimization parameter.Finally, we establish that Dynamic Dominating Set becomes W[2]W[2]-hard when the target solution is required not to be larger than the initial one, even if the edit parameter is exactly one.