A General Framework for Dynamic Consistent Submodular Maximization

2026-06-03

AI Summary

This work addresses the fully dynamic submodular maximization problem, which supports both insertions and deletions, by proposing the first general algorithmic framework that achieves constant-factor approximation guarantees with sublinear adaptive complexity. Built upon a dynamic streaming model, the framework integrates greedy strategies with a buffering mechanism and applies to both cardinality and rank-$k$ matroid constraints. Specifically, under a cardinality constraint, it attains a $(1/2 - O(\varepsilon))$-approximation with $O(1/\varepsilon^2)$ adaptivity; under a rank-$k$ matroid constraint, it achieves a $(1/4 - O(\varepsilon))$-approximation with $O(\log k / \varepsilon^2)$ adaptivity. This represents a significant advance over prior approaches, which were limited to insertion-only settings.

Abstract

Consistency is an important property in dynamic submodular maximization and entails maintaining a near-optimal solution at all times, making only a small number of adjustments to the solution in each step. Prior work has explored this question for the insertion-only case, where the algorithm faces a stream of $n$ insertions, and has established lower and upper bounds for the cardinality-constrained version of the problem. We consider this question in the fully dynamic setting, where the stream of operations may contain both insertions and deletions. We develop a general framework for designing algorithms for this setting, and instantiate it to obtain the first constant-factor approximations with sublinear consistency. For cardinality constraints, we propose a $\frac 12 - O(\varepsilon)$ approximation that is $O\left(\frac{1}{\varepsilon^2}\right)$ consistent. For rank-$k$ matroid constraints, we construct a $\frac 14 - O(\varepsilon)$ approximation to the dynamic optimum that is $O\left(\frac{\log k}{\varepsilon^2}\right)$ consistent.

Problem

• dynamic submodular maximization
• consistency
• insertions and deletions
• cardinality constraints
• matroid constraints

Innovation

• dynamic submodular maximization
• consistency
• fully dynamic setting
• constant-factor approximation
• matroid constraints