Kernels for deletion to classes of acyclic digraphs

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 a (directed) pumpkin (Pumpkin Vertex Deletion Set). Their objective was to shed some light on the kernelization complexity of the DFVS problem, a well known open problem in the area of Parameterized Complexity. In this article, we improve the kernel sizes of Out-Forest Vertex Deletion Set from O(k 3 ) to O(k 2 ) and of Pumpkin Vertex Deletion Set from O(k 18 ) to O(k 3 ). We also prove that the former kernel size is tight under certain complexity theoretic assumptions.

A Polynomial Kernel for Funnel Arc Deletion Set

“…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].…”

Faster parameterized algorithm for pumpkin vertex deletion set

“…Agrawal et al [2] gave an O * (2.562 k )-time algorithm for PVDS. Polynomial kernels for this problem were given in [1,4].…”

“…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).…”

“…). 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).…”

“…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.…”