Dynamic Constrained Submodular Optimization with Polylogarithmic Update Time

Maximizing a monotone submodular function under cardinality constraint k is a core problem in machine learning and database with many basic applications, including video and data summarization, recommendation systems, feature extraction, exemplar clustering, and coverage problems. We study this classic problem in the fully dynamic model where a stream of insertions and deletions of elements of an underlying ground set is given and the goal is to maintain an approximate solution using a fast update time. A recent paper at NeurIPS'20 by Lattanzi, Mitrovic, Norouzi-Fard, Tarnawski, Zadimoghaddam [23] claims to obtain a dynamic algorithm for this problem with a ( 1 2ǫ) approximation ratio and a query complexity bounded by poly(log(n), log(k), ǫ -1 ). However, as we explain in this paper, the analysis has some important gaps. Having a dynamic algorithm for the problem with polylogarithmic update time is even more important in light of a recent result by Chen and Peng [11] at STOC'22 who show a matching lower bound for the problem -any randomized algorithm with a 1 2 + ǫ approximation ratio must have an amortized query complexity that is polynomial in n. In this paper, we develop a simpler algorithm for the problem that maintains a ( 1 2ǫ)-approximate solution for submodular maximization under cardinality constraint k using a polylogarithmic amortized update time. * Appears in ICML'23. An earlier version of this paper was submitted to the SODA'23 conference in July 2022. In February 2023, the authors of [23] contacted us to mention that one of the authors was a referee for our SODA'23 submission, they agree with a bug and they will have a fix for it in their revised arxiv paper [24] .