Distributed strategy selection: A submodular set function maximization approach

Abstract

Joint utility-maximization problems for multi-agent systems often should be addressed by distributed strategy-selection formulation. Constrained by discrete feasible strategy sets, these problems are broadly formulated as NP-hard combinatorial optimization problems. In many cases, these problems can be cast as constrained submodular set function maximization problems, which also belong to the NP-hard domain of problems. A prominent example is the problem of multi-agent mobile sensor dispatching over a discrete domain. This paper considers a class of submodular optimization problems that consist of maximization of a monotone and submodular set function subject to a partition matroid constraint over a group of networked agents that communicate over a connected undirected graph. We work with the value oracle model. Consequently, the only access of the agents to the utility function is through a black box that returns the utility function value given a specific strategy set. We propose a distributed suboptimal polynomial-time algorithm that enables each agent to obtain its respective strategy via local interactions with its neighboring agents. Our solution is a fully distributed gradient-based algorithm using the submodular set functions’ multilinear extension followed by a distributed stochastic Pipage rounding procedure. This algorithm results in a strategy set that when the team utility function is evaluated at the worst case, the utility function value is in 1c(1−e−c−O(1/T)) of the optimal solution with c being the curvature of the submodular function. An example demonstrates our results.