Fully Dynamic Algorithm for Constrained Submodular Optimization

TL;DR

This paper introduces a randomized fully dynamic algorithm for constrained submodular maximization, achieving near-half approximation with efficient updates, applicable to various machine learning tasks.

Contribution

It presents the first fully dynamic algorithm with poly-logarithmic update time for constrained submodular maximization, combining theoretical guarantees with empirical validation.

Findings

Achieves a $(1/2- ext{epsilon})$-approximate solution with efficient updates.

Demonstrates strong empirical performance on real datasets.

Provides theoretical analysis of the algorithm's complexity and approximation ratio.

Abstract

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 $(1/2-\epsilon)$-approximate solution. We complement our theoretical analysis with an empirical study of the performance of our algorithm.