Title: A Universal Framework for Maintaining Dynamic Consistency in Submodular Maximization

Abstract: In the realm of dynamic submodular maximization, consistency is a critical characteristic, defined by the ability to sustain a solution that remains near-optimal throughout the process, requiring only minimal modifications at each step. Previous research has primarily addressed the insertion-only scenario, wherein the algorithm processes a sequence of $n$ insertions, successfully deriving both upper and lower bounds for the cardinality-constrained variant. This study extends the inquiry to the fully dynamic environment, where the input stream comprises both insertions and deletions. We introduce a comprehensive framework for algorithm design within this context and demonstrate its application to achieve the first constant-factor approximations featuring sublinear consistency. Specifically, under cardinality constraints, we present a $\frac 12 - O(\varepsilon)$ approximation algorithm that exhibits $O\left(\frac{1}{\varepsilon^2}\right)$ consistency. Furthermore, for rank-$k$ matroid constraints, we devise a $\frac 14 - O(\varepsilon)$ approximation relative to the dynamic optimum, which maintains $O\left(\frac{\log k}{\varepsilon^2}\right)$ consistency.

Source: arXiv Generated at: 2026-06-04 00:00:00 UTC