מירב זהבי

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 such that F = D − S is an acyclic digraph. In a recent paper, Mnich and van Leeuwen [STACS 2016] studied the kernelization complexity of DFVS with an additional restriction on F—namely that F must be an out-forest, an out-tree, or a (directed) pumpkin—with an objective of shedding some light on the kernelization complexity of the DFVS problem, a well known open problem in the area. The vertex deletion problems corresponding to obtaining an out-forest, an out-tree, or a (directed) pumpkin are Out-forest/Out-tree/Pumpkin Vertex Deletion Set, respectively. They showed that Out-forest/Out-tree/Pumpkin Vertex Deletion Set admit polynomial kernels. Another open problem regarding DFVS is that, does DFVS admit an algorithm with running time 2 ^{O ( k )}n^{O ( 1 )}? We complement the kernelization programme of Mnich and van Leeuwen by designing fast FPT algorithms for the above mentioned problems. In particular, we design an algorithm for Out-forest Vertex Deletion Set that runs in time O(2.73 2 ^kn^{O ( 1 )}) and algorithms for Pumpkin/Out-tree Vertex Deletion Set that runs in time O(2.56 2 ^kn^{O ( 1 )}). As a corollary of our FPT algorithms and the recent result of Fomin et al. [STOC 2016] which gives a relation between FPT algorithms and exact algorithms, we get exact algorithms for Out-forest/Out-tree/Pumpkin Vertex Deletion Set that run in time O(1.63 3 ⁿn^{O ( 1 )}) , O(1.60 9 ⁿn^{O ( 1 )}) and O(1.60 9 ⁿn^{O ( 1 )}) , respectively.