Distributed Submodular Minimization over Networks: a Greedy Column Generation Approach
It provides a fully distributed solution for submodular minimization in unreliable networks, a problem relevant to multi-agent systems and machine learning.
The paper proposes a distributed algorithm for submodular minimization over asynchronous, unreliable, time-varying directed networks, where each agent only knows the function for subsets containing itself. The algorithm converges in finite time to an optimal solution, demonstrated on s-t minimum graph cut instances.
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.