Abstract
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 feasible solution is bounded by some increment-parameter. Parameterized versions of a number of dynamic problems are studied where the solution to the base problem is assumed to be connected. We show that connectivity of solutions plays a positive role with respect to the edit-parameter by proving that the dynamic versions of Connected Dominating Set and Connected Vertex Cover are fixed-parameter tractable with respect to the edit-parameter. On the other hand, the two problems are shown to be \(W[2]\)-hard with respect to the increment-parameter. We illustrate further the utility of connected solutions by proving that Dynamic Independent Dominating Set is \(W[2]\)-hard with respect to the edit-parameter and we discuss some dynamic versions of maximization problems.
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Abu-Khzam, F.N., Egan, J., Fellows, M.R., Rosamond, F.A., Shaw, P. (2014). On the Parameterized Complexity of Dynamic Problems with Connectivity Constraints. In: Zhang, Z., Wu, L., Xu, W., Du, DZ. (eds) Combinatorial Optimization and Applications. COCOA 2014. Lecture Notes in Computer Science(), vol 8881. Springer, Cham. https://doi.org/10.1007/978-3-319-12691-3_47
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DOI: https://doi.org/10.1007/978-3-319-12691-3_47
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