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

2025-11-04

AI Summary

This paper studies the maximization of non-monotone DR-submodular functions under a cardinality constraint $k$. For ground sets of size $n$, we propose two efficient approximation algorithms—FastDrSub and FastDrSub++—that achieve, for the first time, constant-factor approximations with $O(n log k)$ function evaluation complexity: FastDrSub++ attains a $(1/4 - varepsilon)$-approximation guarantee. Our methods integrate a greedy framework with randomized sampling to substantially reduce query overhead while preserving theoretical performance bounds. Experiments on canonical tasks—including influence maximization—demonstrate that our algorithms outperform state-of-the-art approaches in both solution quality and computational efficiency, achieving superior scalability and practicality.

Technology Category

• Search and Optimization: Sampling/Simulation-based Search
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking

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.

Problem

• Maximizing non-monotone DR-submodular functions under size constraints
• Developing efficient approximation algorithms with low query complexity
• Improving solution quality for revenue maximization problems

Innovation

• FastDrSub algorithm with O(n log k) queries
• FastDrSub++ achieves 1/4-ε approximation ratio
• First constant-ratio approximation with low complexity