A General Framework for Dynamic Consistent Submodular Maximization
This work addresses the problem of maintaining near-optimal solutions in fully dynamic submodular maximization, providing the first algorithms with provable guarantees for consistency in the presence of deletions.
The paper develops a general framework for dynamic submodular maximization with both insertions and deletions, achieving the first constant-factor approximations with sublinear consistency: a 1/2 - O(ε) approximation with O(1/ε²) consistency for cardinality constraints, and a 1/4 - O(ε) approximation with O(log k/ε²) consistency for rank-k matroid constraints.
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.