A Dynamic Algorithm for Submodular Maximization with a Knapsack Constraint

Submodular maximization is a fundamental problem in computer science theory, as well as occupies a significant place in machine learning and data mining applications, including influence maximization in social networks, recommendation systems, exemplar-based clustering, viral marketing, and data summarization. In this paper, we study this classic problem under a knapsack constraint in a fully dynamic setting. Given a stream of arbitrary interleaved element insertions and deletions, the dynamic setting requires a solution quickly after each update. By adopting repeated sampling, threshold greedy and filtering technique, our algorithm achieves an approximation ratio of 1/3 for monotone submodular objective function, with polylogarithmic amortized update time. We extensively evaluate the performance of our algorithm against the existing work in two applications including maximum coverage and movie recommendation. The experimental results demonstrate the superiority of our algorithm.