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 ] -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 ] -hard with respect to the increment-parameter. Moreover, we show that Dynamic Independent Dominating Set is 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 ] -hard if the increment-parameter is replaced with the reoptimization parameter. Finally, we establish that Dynamic Dominating Set becomes 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.

Authors

• Faisal N. Abu-Khzam
• Judith Egan
• Michael R. Fellows
• Frances A. Rosamond
• Peter Shaw

Keywords

• Dynamic problems
• Re-optimization
• Dynamic Connected Dominating Set
• Parameterized complexity