Scalable and distributed submodular maximization with matroid constraints

Abstract

Submodular maximization enables efficient approximation of machine learning, networking, and language processing problems. Typically, these problems have been shown to have matroid constraints, which generalize matching and partition conditions. Developing scalable, distributed submodular optimization algorithms that guarantee the same performance as centralized techniques has been an active area of research. In this paper, we address the problem of developing scalable distributed algorithms for submodular maximization with a matroid constraint. Our key step is to construct an auxiliary function from the submodular objective function, and develop distributed exchange-based algorithms for optimizing the auxiliary function. We first introduce a distributed algorithm for maximizing a submodular function with a matroid constraint. We then develop an algorithm for maximizing time-varying submodular functions under partition matroid constraints, which arises in sensor placement and data caching. We prove that both algorithms provide (1−1/e) optimality bounds, and hence achieve the same guarantees as the best centralized algorithms.