Fully Dynamic Submodular Maximization over Matroids
Paul Dütting, Federico Fusco, Silvio Lattanzi, Ashkan Norouzi-Fard, Morteza Zadimoghaddam
TL;DR
This work addresses fully dynamic submodular maximization under a matroid constraint by introducing a randomized algorithm with amortized update time $\tilde{O}(k^2)$ that maintains a solution achieving a constant-factor approximation to the current optimum. The approach centers on a level-based data structure that emulates the Swapping algorithm, reordering and delaying low-robustness elements to handle deletions efficiently while reducing matroid independence queries. The authors prove a deterministic $1/4$-approximation guarantee relative to the dynamic optimum for every update and provide a tight running-time analysis, including a doubling trick to remove the need for pre-knowledge of the stream length and a Delta-free refinement via parallel thresholding achieving $(4+\varepsilon)$-approximation with trade-offs. This work advances the state of fully dynamic submodular optimization under general matroid constraints and opens avenues for further optimization in update time, worst-case versus average-case behavior, and non-adversarial dynamic models.
Abstract
Maximizing monotone submodular functions under a matroid constraint is a classic algorithmic problem with multiple applications in data mining and machine learning. We study this classic problem in the fully dynamic setting, where elements can be both inserted and deleted in real-time. Our main result is a randomized algorithm that maintains an efficient data structure with an $\tilde{O}(k^2)$ amortized update time (in the number of additions and deletions) and yields a $4$-approximate solution, where $k$ is the rank of the matroid.