Thresholding Methods for Streaming Submodular Maximization with a Cardinality Constraint and Its Variants

Abstract

Constrained submodular maximization (CSM) is widely used in numerous data mining and machine learning applications such as data summarization, network monitoring, exemplar-clustering, and nonparametric learning. The CSM can be described as: Given a ground set, a specified constraint, and a submodular set function defined on the power set of the ground set, the goal is to select a subset that satisfies the constraint such that the function value is maximized. Generally, the CSM is NP-hard, and cardinality constrained submodular maximization is well researched. The greedy algorithm and its variants have good performance guarantees for constrained submodular maximization. When dealing with large input scenario, it is usually formulated as streaming constrained submodular maximization (SCSM), and the classical greedy algorithm is usually inapplicable. The streaming model uses a limited memory to extract a small fraction of items at any given point of time such that the specified constraint is satisfied, and good performance guarantees are also maintained. In this chapter, we list the up-to-date popular algorithms for streaming submodular maximization with cardinality constraint and its variants, and summarize some problems in streaming submodular maximization that are still open.