Matrix Scaling: a New Heuristic for the Feedback Vertex Set Problem

For a digraph $G$, a set $F\subseteq V(G)$ is said to be a feedback vertex set (FVS) if $G-F$ is acyclic. The problem of finding a smallest FVS is NP-hard. We present a matrix scaling technique for finding feedback vertex sets in un-weighted directed graphs that runs in $O(|F|\log(|V|)|V|^{2})$ time. Our technique is empirically shown to produce smaller feedback vertex sets than other known heuristics and in a shorter amount of time.