Extremal results on feedback arc sets in digraphs

Abstract

For an oriented graph (Formula presented.), let (Formula presented.) denote the size of a minimum feedback arc set, a smallest edge subset whose deletion leaves an acyclic subgraph. Berger and Shor proved that any (Formula presented.) -edge oriented graph (Formula presented.) satisfies (Formula presented.). We observe that if an oriented graph (Formula presented.) has a fixed forbidden subgraph (Formula presented.), the bound (Formula presented.) is sharp as a function of (Formula presented.) if (Formula presented.) is not bipartite, but the exponent (Formula presented.) in the lower order term can be improved if (Formula presented.) is bipartite. Using a result of Bukh and Conlon on Turán numbers, we prove that any rational number in (Formula presented.) is optimal as an exponent for some finite family of forbidden subgraphs. Our upper bounds come equipped with randomized linear-time algorithms that construct feedback arc sets achieving those bounds. We also characterize directed quasirandomness via minimum feedback arc sets.