Affiliation:
1. Department of Mathematics Stanford University Stanford California USA
2. Department of Mathematics Columbia University New York New York USA
3. Department of Mathematics Massachusetts Institute of Technology Cambridge Massachusetts USA
Abstract
AbstractFor an oriented graph , let 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 ‐edge oriented graph satisfies . We observe that if an oriented graph has a fixed forbidden subgraph , the bound is sharp as a function of if is not bipartite, but the exponent in the lower order term can be improved if is bipartite. Using a result of Bukh and Conlon on Turán numbers, we prove that any rational number in 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.
Funder
National Science Foundation
Subject
Applied Mathematics,Computer Graphics and Computer-Aided Design,General Mathematics,Software