  Submodular optimization is a special class of combinatorial optimization
arising in several machine learning problems, but also in cooperative control
of complex systems. In this paper, we consider agents in an asynchronous,
unreliable and time-varying directed network that aim at cooperatively solving
submodular minimization problems in a fully distributed way. The challenge is
that the (submodular) objective set-function is only partially known by agents,
that is, each one is able to evaluate the function only for subsets including
itself. We propose a distributed algorithm based on a proper linear programming
reformulation of the combinatorial problem. Our algorithm builds on a column
generation approach in which each agent maintains a local candidate basis and
locally generates columns with a suitable greedy inner routine. A key
interesting feature of the proposed algorithm is that the pricing problem,
which involves an exponential number of constraints, is solved by the agents
through a polynomial time greedy algorithm. We prove that the proposed
distributed algorithm converges in finite time to an optimal solution of the
submodular minimization problem and we corroborate the theoretical results by
performing numerical computations on instances of the $s$--$t$ minimum graph
cut problem.
