Abstract:In the Directed Feedback Vertex Set (DFVS) problem, we are given a digraph D on n vertices and a positive integer k and the objective is to check whether there exists a set of vertices S of size at most k such that F = D − S is a directed acyclic digraph. In a recent paper, Mnich and van Leeuwen [STACS 2016 ] considered the kernelization complexity of DFVS with an additional restriction on F , namely that F must be an out-forest (Out-Forest Vertex Deletion Set), an out-tree (Out-Tree Vertex Deletion Set), or … Show more
“…Out-Forest-ADS and Pumpkin-ADS can be solved in polynomial time [12], while Out-Forest-VDS and Pumpkin-VDS are NP-hard and admit polynomial kernels [2; 12] of size O(k 2 ) and O(k 3 ), respectively [2]. F η -VDS admits a polynomial kernel for constant η, where F η is the class of all digraphs with (undirected) treewidth at most η [10].…”
In Directed Feedback Arc Set (DFAS) we search for a set of at most k arcs which intersect every cycle in the input digraph. It is a wellknown open problem in parameterized complexity to decide if DFAS admits a kernel of polynomial size. We consider C-Arc Deletion Set (C-ADS), a variant of DFAS where we want to remove at most k arcs from the input digraph in order to turn it into a digraph of a class C. In this work, we choose C to be the class of funnels. Funnel-ADS is NP-hard even if the input is a DAG, but is fixed-parameter tractable with respect to k. So far no polynomial kernels for this problem were known. Our main result is a kernel for Funnel-ADS with O(k 6 ) many vertices and O(k 7 ) many arcs, computable in linear time.
“…Out-Forest-ADS and Pumpkin-ADS can be solved in polynomial time [12], while Out-Forest-VDS and Pumpkin-VDS are NP-hard and admit polynomial kernels [2; 12] of size O(k 2 ) and O(k 3 ), respectively [2]. F η -VDS admits a polynomial kernel for constant η, where F η is the class of all digraphs with (undirected) treewidth at most η [10].…”
In Directed Feedback Arc Set (DFAS) we search for a set of at most k arcs which intersect every cycle in the input digraph. It is a wellknown open problem in parameterized complexity to decide if DFAS admits a kernel of polynomial size. We consider C-Arc Deletion Set (C-ADS), a variant of DFAS where we want to remove at most k arcs from the input digraph in order to turn it into a digraph of a class C. In this work, we choose C to be the class of funnels. Funnel-ADS is NP-hard even if the input is a DAG, but is fixed-parameter tractable with respect to k. So far no polynomial kernels for this problem were known. Our main result is a kernel for Funnel-ADS with O(k 6 ) many vertices and O(k 7 ) many arcs, computable in linear time.
“…Agrawal et al [2] gave an O * (2.562 k )-time algorithm for PVDS. Polynomial kernels for this problem were given in [1,4].…”
Section: Introductionmentioning
confidence: 99%
“…Therefore, S contains at least 3 vertices of {w 1 , w 2 , w 3 , w 4 }. The branching vector of Rule ( 3) is (1,3,3,3,3).…”
mentioning
confidence: 99%
“…). In the first case we have that N − (w 1 ) \ {v} ⊆ S (otherwise d − G−S (w 1 ) ≥ 2) and in the second case N − (w 2 ) \ {v} ⊆ S. The branching vector of Rule (6) is at least (1,3,3,2).…”
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confidence: 99%
“…The branching vectors of the branching rules of the algorithm are (1, 1), (1, 2, 2), (1,3,3,2), and (1, 3, 3, 3, 3) (in the worst cases). All these vectors have branching number 2.…”
In the Cluster Vertex Deletion problem the input is a graph G and an integer k. The goal is to decide whether there is a set of vertices S of size at most k such that the deletion of the vertices of S from G results a graph in which every connected component is a clique. We give an algorithm for Cluster Vertex Deletion whose running time is O * (1.811 k ).
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