Fast Approximation Algorithm for Non-Monotone DR-submodular Maximization under Size Constraint

Abstract

This work studies the non-monotone DR-submodular Maximization over a ground set of $n$ subject to a size constraint $k$. We propose two approximation algorithms for solving this problem named FastDrSub and FastDrSub++. FastDrSub offers an approximation ratio of $0.044$ with query complexity of $O(n\log (k))$. The second one, FastDrSub++, improves upon it with a ratio of $1/4-\hat{I}\mu$ within query complexity of $(n\log k)$ for an input parameter $\hat{I}\mu >0$. Therefore, our proposed algorithms are the first constant-ratio approximation algorithms for the problem with the low complexity of $O(n\log (k))$. Additionally, both algorithms are experimentally evaluated and compared against existing state-of-the-art methods, demonstrating their effectiveness in solving the Revenue Maximization problem with DR-submodular objective function. The experimental results show that our proposed algorithms significantly outperform existing approaches in terms of both query complexity and solution quality.