Optimal Algorithms for Submodular Maximization with Distributed Constraints

TL;DR

This paper introduces a distributed continuous greedy algorithm for maximizing submodular functions under distributed constraints, achieving the optimal approximation ratio with local computation and communication.

Contribution

It develops the first distributed algorithm that attains the tight (1-1/e) approximation for submodular maximization with distributed constraints, surpassing previous greedy methods.

Findings

The proposed CDCG algorithm achieves the (1-1/e) approximation ratio.

Empirical results show CDCG outperforms sequential greedy in multi-agent coverage.

The method effectively combines continuous relaxation with distributed consensus.

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.