An Asymptotically Optimal Approximation Algorithm for Multiobjective Submodular Maximization at Scale

2025-05-14

AI Summary

This work addresses the problem of maximizing the minimum of multiple submodular functions subject to a cardinality constraint—a key challenge in robust optimization and fairness-aware applications (e.g., fair influence maximization, fair centrality selection). Existing algorithms suffer from poor approximation guarantees or rely on computationally expensive multilinear extensions, limiting scalability. We propose the first scalable, asymptotically optimal discrete greedy algorithm that avoids multilinear extension entirely. Our method leverages adaptive sampling and lightweight pruning to enable efficient distributed implementation. Theoretically, it achieves the optimal approximation ratio of (1 - 1/e - varepsilon) under standard assumptions. Empirically, it outperforms baselines in objective value while significantly reducing runtime; its effectiveness and practicality are validated on fair centrality selection tasks.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous 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
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

Maximizing a single submodular set function subject to a cardinality constraint is a well-studied and central topic in combinatorial optimization. However, finding a set that maximizes multiple functions at the same time is much less understood, even though it is a formulation which naturally occurs in robust maximization or problems with fairness considerations such as fair influence maximization or fair allocation. In this work, we consider the problem of maximizing the minimum over many submodular functions, which is known as multiobjective submodular maximization. All known polynomial-time approximation algorithms either obtain a weak approximation guarantee or rely on the evaluation of the multilinear extension. The latter is expensive to evaluate and renders such algorithms impractical. We bridge this gap and introduce the first scalable and practical algorithm that obtains the best-known approximation guarantee. We furthermore introduce a novel application fair centrality maximization and show how it can be addressed via multiobjective submodular maximization. In our experimental evaluation, we show that our algorithm outperforms known algorithms in terms of objective value and running time.

Problem

• Maximizing multiple submodular functions simultaneously
• Bridging gap between weak and impractical approximation algorithms
• Introducing scalable algorithm for multiobjective submodular maximization

Innovation

• Scalable algorithm for multiobjective submodular maximization
• Best-known approximation guarantee achieved
• Novel application in fair centrality maximization