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

TL;DR

This paper introduces two fast approximation algorithms, FastDrSub and FastDrSub++, for non-monotone DR-submodular maximization under size constraints, achieving the first constant-ratio guarantees with low query complexity.

Contribution

The paper presents the first constant-ratio approximation algorithms with low complexity for non-monotone DR-submodular maximization under size constraints.

Findings

FastDrSub achieves a 0.044 approximation ratio with O(n log k) queries.

FastDrSub++ improves the ratio to 1/4 - ε with similar query complexity.

Experimental results show significant improvements over existing methods in solution quality and efficiency.

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 $1/4-\epsilon$ within query complexity of $(n \log k)$ for an input parameter $\epsilon >0$. 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.