Multi-Agent Maximization of a Monotone Submodular Function via Maximum Consensus

This paper studies distributed submodular optimization subject to partition matroid. We work in the value oracle model where the only access of the agents to the utility function is through a black box that returns the utility function value. The agents are communicating over a connected undirected graph and have access only to their own strategy set. As known in the literature, submodular maximization subject to matroid constraints is NP-hard. Hence, our objective is to propose a polynomial-time distributed algorithm to obtain a suboptimal solution with guarantees on the optimality bound. Our proposed algorithm is based on a distributed stochastic gradient ascent scheme built on the multilinear-extension of the submodular set function. We use a maximum consensus protocol to minimize the inconsistency of the shared information over the network caused by delay in the flow of information while solving for the fractional solution of the multilinear extension model. Furthermore, we propose a distributed framework of finding a set solution using the fractional solution. We show that our distributed algorithm results in a strategy set that when the team objective function is evaluated at worst case the objective function value is in 1−1/e−O(1/T) of the optimal solution in the value oracle model where T is the number of communication instances of the agents. An example demonstrates our results.