Optimal Algorithms for Submodular Maximization with Distributed Constraints

• Main Authors: Robey, Alexander, Adibi, Arman, Schlotfeldt, Brent, Pappas, George J, Hassani, Hamed
• Format: Journal Article
• Language: English
• Published: 30.09.2019
• Subjects: Computer Science - Data Structures and Algorithms; Computer Science - Learning; Mathematics - Optimization and Control; Statistics - Machine Learning
• DOI: 10.48550/arxiv.1909.13676

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.