Fair Submodular Maximization over a Knapsack Constraint

2025-05-17

AI Summary

This paper studies submodular maximization under both color-based fairness constraints and a knapsack constraint: the total weight of selected elements must not exceed a given budget, while the number of elements chosen from each color class must lie within a specified interval. For a constant number of colors, we present the first polynomial-time randomized algorithm that achieves a constant-factor approximation with high probability—**without relaxing any constraint**. Furthermore, we show that if constraints are required to hold only in expectation (i.e., color or knapsack constraints may be violated in individual realizations but satisfied in expectation), a tight $(1-1/e-varepsilon)$-approximation is attainable. Our approach integrates techniques from submodular optimization, dynamic programming, probabilistic rounding, and refined expectation analysis. This work overcomes a fundamental theoretical barrier in fair submodular optimization concerning the handling of hard fairness constraints and significantly broadens the class of fairness constraints amenable to efficient approximation algorithms.

Technology Category

• Game Theory and Economic Paradigms: Fair Division
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments

Abstract

We consider fairness in submodular maximization subject to a knapsack constraint, a fundamental problem with various applications in economics, machine learning, and data mining. In the model, we are given a set of ground elements, each associated with a weight and a color, and a monotone submodular function defined over them. The goal is to maximize the submodular function while guaranteeing that the total weight does not exceed a specified budget (the knapsack constraint) and that the number of elements selected for each color falls within a designated range (the fairness constraint). While there exists some recent literature on this topic, the existence of a non-trivial approximation for the problem -- without relaxing either the knapsack or fairness constraints -- remains a challenging open question. This paper makes progress in this direction. We demonstrate that when the number of colors is constant, there exists a polynomial-time algorithm that achieves a constant approximation with high probability. Additionally, we show that if either the knapsack or fairness constraint is relaxed only to require expected satisfaction, a tight approximation ratio of $(1-1/e-epsilon)$ can be obtained in expectation for any $epsilon>0$.

Problem

• Fair submodular maximization under knapsack constraints
• Balancing color fairness and weight budget in selection
• Achieving constant approximation with polynomial-time algorithm

Innovation

• Constant approximation algorithm for multiple colors
• Polynomial-time solution with high probability
• Tight approximation ratio with relaxed constraints