Parameterized Complexity of Critical Node Cuts

Danny Hermelin; Moshe Kaspi; Christian Komusiewicz; Barak Navon

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arXiv · 1503.06321

Parameterized Complexity of Critical Node Cuts

Danny Hermelin·Moshe Kaspi·Christian Komusiewicz·Barak Navon

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Abstract

We consider the following natural graph cut problem called Critical Node Cut (CNC): Given a graph $G$ on $n$ vertices, and two positive integers $k$ and $x$, determine whether $G$ has a set of $k$ vertices whose removal leaves $G$ with at most $x$ connected pairs of vertices. We analyze this problem in the framework of parameterized complexity. That is, we are interested in whether or not this problem is solvable in $f(\kappa) \cdot n^{O(1)}$ time (i.e., whether or not it is fixed-parameter tractable), for various natural parameters $\kappa$. We consider four such parameters: - The size $k$ of the required cut. - The upper bound $x$ on the number of remaining connected pairs. - The lower bound $y$ on the number of connected pairs to be removed. - The treewidth $w$ of $G$. We determine whether or not CNC is fixed-parameter tractable for each of these parameters. We determine this also for all possible aggregations of these four parameters, apart from $w+k$. Moreover, we also determine whether or not CNC admits a polynomial kernel for all these parameterizations. That is, whether or not there is an algorithm that reduces each instance of CNC in polynomial time to an equivalent instance of size $\kappa^{O(1)}$, where $\kappa$ is the given parameter.

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BibTeX RIS

Danny Hermelin, Moshe Kaspi, Christian Komusiewicz, Barak Navon. 2015-03-21. Parameterized Complexity of Critical Node Cuts. https://arxiv.org/abs/1503.06321

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