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

TL;DR

This paper introduces a distributed, gradient-based algorithm for maximizing a monotone submodular function under a matroid constraint in multi-agent systems, achieving near-optimal solutions through local interactions.

Contribution

It proposes a novel fully distributed algorithm using multilinear extension and maximum consensus for submodular maximization in multi-agent networks.

Findings

Achieves a solution within 1-1/e-O(1/T) of the optimal.

Operates in polynomial time with local communications.

Demonstrated effectiveness through an example.

Abstract

Constrained submodular set function maximization problems often appear in multi-agent decision-making problems with a discrete feasible set. A prominent example is the problem of multi-agent mobile sensor placement over a discrete domain. However, submodular set function optimization problems are known to be NP-hard. In this paper, we consider a class of submodular optimization problems that consists of maximization of a monotone and submodular set function subject to a uniform matroid constraint over a group of networked agents that communicate over a connected undirected graph. Our objective is to obtain a distributed suboptimal polynomial-time algorithm that enables each agent to obtain its respective policy via local interactions with its neighboring agents. Our solution is a fully distributed gradient-based algorithm using the multilinear extension of the submodular set functions and exploiting a maximum consensus scheme. This algorithm results in a policy 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. An example demonstrates our results.