Asymptotic Lower Bounds for the Feedback Arc Set Problem in Random Graphs

Given a directed graph, the Minimum Feedback Arc Set (FAS) problem asks for a minimum (size) set of arcs in a directed graph, which, when removed, results in an acyclic graph. In a seminal paper, Berger and Shor [1], in 1990, developed initial upper bounds for the FAS problem in general directed graphs. Here we find asymptotic \textit{lower bounds} for the FAS problem in a class of random, oriented, directed graphs derived from the Erd\H{o}s-R\'{e}nyi model $G(n,M)$, with n vertices and M (undirected) edges, the latter randomly chosen. Each edge is then randomly given a direction to form our directed graph. We show that $$Pr\left(\textbf{Y}^* \le M \left( \frac{1}{2} -\sqrt{\frac{\log n}{\Delta_{av}}}\right)\right)$$ approaches zero exponentially in $n$, with $\textbf{Y}^*$ the (random) size of the minimum feedback arc set and $\Delta_{av}=2M/n$ the average vertex degree. Lower bounds for random tournaments, a special case, were obtained by Spencer [12] and de la Vega [13] and these are discussed. In comparing the bound above to averaged experimental FAS data on related random graphs developed by K. Hanauer [7] we find that the approximation $\textbf{Y}^*_{av} \approx M\left( \frac{1}{2} -\frac{1}{2}\sqrt{\frac{\log n}{\Delta_{av}}}\right)$ lies remarkably close graphically to the algorithmically computed average size $\textbf{Y}^*_{av}$ of minimum feedback arc sets.