Tight upper bounds for minimum feedback arc sets of regular graphs

Abstract

The Feedback Arc Set problem is one of the classical NP-hard problems. Given a graph with n vertices and m arcs, it asks for a subset of arcs whose deletion makes a graph acyclic. An equivalent is the Linear Ordering, where the vertices are ordered from 1 to n, and a feedback arc is an arc that is directed contrarily. Both problems have been studied intensely. Here, we add a new point of view. We first derive properties of linear orderings, that can be established efficiently. Our main result are upper bounds on the cardinality of a minimum feedback arc set for graphs with degree at most 3 and 4. We prove that the bounds are at most n/3 and m/3, respectively, and show that both are tight. © 2013 Springer-Verlag.