Abstract
In the Directed Feedback Vertex Set (DFVS) problem, the input is a directed graph D and an integer k. The objective is to determine whether there exists a set of at most k vertices intersecting every directed cycle of D. DFVS was shown to be fixed-parameter tractable when parameterized by solution size by Chen et al. (J ACM 55(5):177–186, 2008); since then, the existence of a polynomial kernel for this problem has become one of the largest open problems in the area of parameterized algorithmics. Since this problem has remained open in spite of the best efforts of a number of prominent researchers and pioneers in the field, a natural step forward is to study the kernelization complexity of DFVS parameterized by a natural larger parameter. In this paper, we study DFVS parameterized by the feedback vertex set number of the underlying undirected graph. We provide two main contributions: a polynomial kernel for this problem on general instances, and a linear kernel for the case where the input digraph is embeddable on a surface of bounded genus.
Highlights
Feedback Set problems are fundamental combinatorial optimization problems
For over a decade resolving the fixed-parameter tractability of Directed Feedback Vertex Set (DFVS) (whether there is an algorithm running in time f (k) · nO(1) for some computable function f ) was considered the most important open problem in parameterized complexity
We show that a set S of size at most k is a DFVS for D if and only if it is a DFVS for D
Summary
Feedback Set problems are fundamental combinatorial optimization problems. Typically, in these problems, we are given a graph G (directed or undirected) and a positive integer k, and the objective is to select at most k vertices, edges or arcs to hit all cycles of the input graph. This was later extended by Cygan et al [13] who systematically studied several generalizations of this problem and obtained several positive as well as negative results with respect to the existence of polynomial kernels While this kind of an alternate parameterization is interesting for problems which are known to have polynomial kernels when parameterized by the solution size, it is the exact opposite approach which is useful when dealing with problems for which this question has been answered negatively or remains open. This paper deals with such a parameterization for DFVS as an intermediate step towards answering the main question, that of a polynomial kernel for DFVS To this end, we chose a natural parameter which is never less than the solution size, in particular the feedback vertex set number of the undirected graph underlying the given digraph. The framework is non-constructive; unlike our Theorem 2, it does not provide a concrete kernelization algorithm for the problem
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