Parameterised Algorithms for Deletion to Classes of DAGs

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 \(2^{\mathcal {O}(k)} n^{\mathcal {O}(1)}\)? 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 \(\mathcal {O}(2.732^{k} n^{\mathcal {O}(1)})\) and algorithms for Pumpkin/Out-tree Vertex Deletion Set that runs in time \(\mathcal {O}(2.562^{k} n^{\mathcal {O}(1)})\). 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 \(\mathcal {O}(1.633^{n} n^{\mathcal {O}(1)})\), \(\mathcal {O}(1.609^{n} n^{\mathcal {O}(1)})\) and \(\mathcal {O}(1.609^{n} n^{\mathcal {O}(1)})\), respectively.