A Contraction Algorithm for finding Minimal Feedback Sets

We present a contraction algorithm for finding a minimal set of arcs or vertices that break all cycles in a digraph. The algorithm is (essentially) an extension to the contraction algorithm for the feedback vertex set problem introduced by Levy and Lew (Levy & Lew 1988). The algorithm’s time complexity of O (m log n) is preserved, while allowing both (weighted) feedback vertices and arcs. As the transformation from feedback arc set to feedback vertex set graph increases the graph’s size, the new algorithm becomes both faster and more powerful than the original one when applied to feedback arc set problems. We will show that the algorithm works well for reducible flow graphs, as it preserves the structure and can easily be combined with the algorithm proposed by Ramachandran (Ramachandran 1988).