TY - JOUR

T1 - Parameterized complexity of critical node cuts

AU - Hermelin, Danny

AU - Kaspi, Moshe

AU - Komusiewicz, Christian

AU - Navon, Barak

N1 - Funding Information: The research leading to these results has received funding from the People Programme (Marie Curie Actions) of the European Union's Seventh Framework Programme ( FP7/2007-2013 ) under REA grant agreement number 631163.11 , by the Israel Science Foundation (Grant No. 1055/14 ). Publisher Copyright: © 2016 Elsevier B.V.

PY - 2016/10/25

Y1 - 2016/10/25

N2 - We consider the following 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(κ)⋅nO(1) time (i.e., whether or not it is fixed-parameter tractable), for various natural parameters κ. 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 κO(1), where κ is the given parameter.

AB - We consider the following 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(κ)⋅nO(1) time (i.e., whether or not it is fixed-parameter tractable), for various natural parameters κ. 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 κO(1), where κ is the given parameter.

KW - Graph cut problems

KW - Kernelization

KW - Treewidth

UR - http://www.scopus.com/inward/record.url?scp=84991373017&partnerID=8YFLogxK

U2 - https://doi.org/10.1016/j.tcs.2016.08.018

DO - https://doi.org/10.1016/j.tcs.2016.08.018

M3 - Article

SN - 0304-3975

VL - 651

SP - 62

EP - 75

JO - Theoretical Computer Science

JF - Theoretical Computer Science

ER -