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

TL;DR

This paper establishes asymptotic lower bounds for the size of minimum feedback arc sets in random directed graphs derived from Erdős-Rényi models, providing theoretical insights and empirical validation.

Contribution

It introduces new asymptotic lower bounds for the feedback arc set problem in random graphs, extending previous work on tournaments and connecting theory with experimental data.

Findings

Lower bounds approach zero exponentially with increasing n.

The derived bounds closely match empirical data.

The results generalize previous bounds for tournaments.

Abstract

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.