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 $O(\mathrm{\mid }F\mathrm{\mid }\log (\mathrm{\mid }V\mathrm{\mid })\mathrm{\mid }V{\mathrm{\mid }}^2)$ time. Our technique is empirically shown to produce smaller feedback vertex sets than other known heuristics and in a shorter amount of time.