Published by the Department of Computer Science, The University of Chicago.
In this note we study the computational complexity of feedback arc set counting problems in directed graphs, highlighting some subtle yet common properties of counting classes. Counting the number of feedback arc sets of cardinality $k$ and the total number of feedback arc sets are $\#P$-complete problems, while counting the number of minimum feedback arc sets is only proven to be $\#P$-hard. Indeed, this latter problem is $\#OptP[\log n]$-complete, hence if it belongs to $\#P$ then $P=NP$.
Submitted Submitted September 29,2020, revised March 10, 2022,published March 26, 2022.