  A number of discrete and continuous optimization problems in machine learning
are related to convex minimization problems under submodular constraints. In
this paper, we deal with a submodular function with a directed graph structure,
and we show that a wide range of convex optimization problems under submodular
constraints can be solved much more efficiently than general submodular
optimization methods by a reduction to a maximum flow problem. Furthermore, we
give some applications, including sparse optimization methods, in which the
proposed methods are effective. Additionally, we evaluate the performance of
the proposed method through computational experiments.
