The path following algorithm was proposed recently to solve the matching problems on undirected graph models, and exhibited a state-of-art performance on matching accuracy. In this paper we extend the path following algorithm to the matching problems on directed graph models, by proposing a concave relaxation for the problem. Based on the concave and convex relaxations, a series of objective functions are constructed, and the Frank-Wolfe algorithm is then utilized to minimize them. Several experiments on synthetic and real data witness the validity of the extended path following algorithm.