Parameterized Complexity of Directed Feedback Set Problems in Tournaments

Abstract

Given a directed graph on n vertices and an integer parameter k, the feedback vertex (arc) set problem asks whether the given graph has a set of k vertices (arcs) whose removal results in an acyclic directed graph. The parameterized complexity of these problems, in the framework introduced by Downey and Fellows, is a long standing open problem in the area. We address these problems in the well studied class of directed graphs called tournaments. While the feedback vertex set problem is easily seen to be fixed parameter tractable in tournaments, we show that the feedback arc set problem is also fixed parameter tractable. Then we address the parametric dual problems (where the k is replaced by ‘all but k’ in the questions) and show that they are fixed parameter tractable in oriented directed graphs (where there is at most one directed arc between a pair of vertices). More specifically, the dual problem we show fixed parameter tractable are: Given an oriented directed graph, is there a subset of k vertices (arcs) that forms an acyclic directed subgraph of the graph?