Fully Dynamic Algorithm for Constrained Submodular Optimization

The task of maximizing a monotone submodular function under a cardinality constraint is at the core of many machine learning and data mining applications, including data summarization, sparse regression and coverage problems. We study this classic problem in the fully dynamic setting, where elements can be both inserted and removed. Our main result is a randomized algorithm that maintains an efficient data structure with a poly-logarithmic amortized update time and yields a ( 1 /2ǫ)-approximate solution. We complement our theoretical analysis with an empirical study of the performance of our algorithm. Version v2. This version fixes correctness issues in the previous version of this result, pointed out by the authors in [BBG + 23]; they also provide a fix. We independently provide another solution here. The main change is that in Peeling, elements are now added one by one instead of in batches, and we explicitly check whether the marginal contribution of every added element is large enough. This implies that our claimed guarantees on the approximation ratio are now deterministic, rather than in expectation; instead, the runtime (oracle complexity) analysis has become more complex. Since the publication of the previous version, in an exciting result, [CP22] have proved that any dynamic algorithm that maintains a better-than-1 /2 approximation must have an amortized query complexity that is polynomial in n.