Optimal Algorithms for Submodular Maximization with Distributed Constraints

Optimal Algorithms for Submodular Maximization with Distributed Constraints Alexander Robey, Arman Adibi, Brent Schlotfeldt, George J. Pappas, and Hamed Hassani Abstract -- We consider a class of discrete optimization problems that aim to maximize a submodular objective function subject to a distributed partition matroid constraint. More precisely, we consider a networked scenario in which multiple agents choose actions from local strategy sets with the goal of maximizing a submodular objective function defined over the set of all possible actions. Given this distributed setting, we develop Constraint-Distributed Continuous Greedy ( CDCG), a message passing algorithm that converges to the tight (1 1 /e) approximation factor of the optimum global solution using only local computation and communication. It is known that a sequential greedy algorithm can only achieve a 1 /2 multiplicative approximation of the optimal solution for this class of problems in the distributed setting. Our framework relies on lifting the discrete problem to a continuous domain and developing a consensus algorithm that achieves the tight (1 1 /e) approximation guarantee of the global discrete solution once a proper rounding scheme is applied. We also offer empirical results from a multi-agent area coverage problem to show that the proposed method significantly outperforms the state-of-the-art sequential greedy method. I. INTRODUCTION Recently, the need has arisen to design algorithms that distribute decision making among a collection of agents or computing devices. This need has been motivated by problems from statistics, machine learning and robotics. These problems include: - (Density estimation) What is the best way to estimate a nonparametric density function from a distributed dataset? Inherent to all of these applications is an underlying optimization problem that can be expressed as maximize f (S) (1a) subject to S Y, S I (1b) where f is a submodular function (i.e. it has a diminishing-returns property), Y is a finite ground set of all decision variables, and I is a family of allowable subsets of Y .