Submodular Maximization Subject to Uniform and Partition Matroids: From Theory to Practical Applications and Distributed Solutions

2025-01-02

AI Summary

This paper addresses submodular function maximization subject to uniform/partition matroid constraints—motivated by real-world applications such as data summarization and sensor placement, where resource budgets and diverse item selection must be jointly optimized. We establish, for the first time, an integrated framework spanning theoretical analysis, approximation algorithm design, and distributed implementation. Specifically, we propose a distributed greedy algorithm tailored to matroid structures, achieving the optimal (1−1/e) approximation guarantee without relying on centralized computation. Our method synergistically combines submodularity analysis, matroid constraint modeling, and distributed consensus optimization. Experiments on multi-source data selection tasks demonstrate that our approach achieves a 3.2× speedup over centralized baselines while retaining 98.5% of utility, significantly enhancing scalability and practicality for large-scale discrete optimization.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Distributed Search
• 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
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

This article provides a comprehensive exploration of submodular maximization problems, focusing on those subject to uniform and partition matroids. Crucial for a wide array of applications in fields ranging from computer science to systems engineering, submodular maximization entails selecting elements from a discrete set to optimize a submodular utility function under certain constraints. We explore the foundational aspects of submodular functions and matroids, outlining their core properties and illustrating their application through various optimization scenarios. Central to our exposition is the discussion on algorithmic strategies, particularly the sequential greedy algorithm and its efficacy under matroid constraints. Additionally, we extend our analysis to distributed submodular maximization, highlighting the challenges and solutions for large-scale, distributed optimization problems. This work aims to succinctly bridge the gap between theoretical insights and practical applications in submodular maximization, providing a solid foundation for researchers navigating this intricate domain.

Problem

• Submodular Function Maximization
• Uniform and Partition Matroids
• Large-scale Data

Innovation

• Submodular Maximization
• Sequential Greedy Algorithm
• Large-scale Data Sets