  A directed graph (digraph) is formed by vertices and arcs (directed edges)
from one vertex to another. A feedback vertex set (FVS) is a set of vertices
that contains at least one vertex of every directed cycle in this digraph. The
directed feedback vertex set problem aims at constructing a FVS of minimum
cardinality. This is a fundamental cycle-constrained hard combinatorial
optimization problem with wide practical applications. In this paper we
construct a spin glass model for the directed FVS problem by converting the
global cycle constraints into local arc constraints, and study this model
through the replica-symmetric (RS) mean field theory of statistical physics. We
then implement a belief propagation-guided decimation (BPD) algorithm for
single digraph instances. The BPD algorithm slightly outperforms the simulated
annealing algorithm on large random graph instances. The predictions of the RS
mean field theory are noticeably lower than the BPD results, possibly due to
its neglect of cycle-caused long range correlations.
