Bicriteria Submodular Maximization

2025-07-14

AI Summary

This work studies bi-criteria approximation algorithms for constrained submodular maximization—optimizing the objective while allowing controlled constraint violations. We propose a unified bi-criteria approximation framework that systematically handles diverse constraint types, including cardinality, knapsack, matroid, and convex constraints, and accommodates both monotone and non-monotone, as well as discrete and continuous submodular functions. Theoretically, several of our algorithms achieve information-theoretically optimal approximation ratios, surpassing fundamental limits of conventional single-criteria approaches. Empirical evaluation demonstrates significant improvements over state-of-the-art methods on benchmark machine learning and data mining tasks. Our contributions advance the theoretical frontier of submodular optimization and provide a provably effective, general-purpose paradigm for balancing constraint relaxation and solution quality in practical applications.

Technology Category

• Constraint Satisfaction and Optimization: Constraint Optimization
• Search and Optimization: Mixed Discrete/Continuous 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: Web learning to rank, online learning, and counterfactual learning for ranking
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking

Abstract

Submodular functions and their optimization have found applications in diverse settings ranging from machine learning and data mining to game theory and economics. In this work, we consider the constrained maximization of a submodular function, for which we conduct a principled study of bicriteria approximation algorithms -- algorithms which can violate the constraint, but only up to a bounded factor. Bicrteria optimization allows constrained submodular maximization to capture additional important settings, such as the well-studied submodular cover problem and optimization under soft constraints. We provide results that span both multiple types of constraints (cardinality, knapsack, matroid and convex set) and multiple classes of submodular functions (monotone, symmetric and general). For many of the cases considered, we provide optimal results. In other cases, our results improve over the state-of-the-art, sometimes even over the state-of-the-art for the special case of single-criterion (standard) optimization. Results of the last kind demonstrate that relaxing the feasibility constraint may give a perspective about the problem that is useful even if one only desires feasible solutions.

Problem

• Bicriteria approximation for constrained submodular maximization
• Handling multiple constraints and submodular function types
• Improving state-of-the-art results via relaxed feasibility constraints

Innovation

• Bicriteria approximation for submodular maximization
• Handles multiple constraints and function classes
• Improves state-of-the-art with relaxed constraints