An Exact Method for the Minimum Feedback Arc Set Problem

A feedback arc set of a directed graph G is a subset of its arcs containing at least one arc of every cycle in G. Finding a feedback arc set of minimum cardinality is an NP-hard problem called the minimum feedback arc set problem. Numerically, the minimum set cover formulation of the minimum feedback arc set problem is appropriate as long as all simple cycles in G can be enumerated. Unfortunately, even those sparse graphs that are important for practical applications often have ω (2n) simple cycles. Here we address precisely such situations: An exact method is proposed for sparse graphs that enumerates simple cycles in a lazy fashion and iteratively extends an incomplete cycle matrix. In all cases encountered so far, only a tractable number of cycles has to be enumerated until a minimum feedback arc set is found. The practical limits of the new method are evaluated on a test set containing computationally challenging sparse graphs, relevant for industrial applications. The 4,468 test graphs are of varying size and density and suitable for testing the scalability of exact algorithms over a wide range. © 2021 ACM.