Group Fair Matchings using Convex Cost Functions

2025-08-17

AI Summary

This paper studies the platform-item allocation problem under group fairness constraints: items are partitioned into attribute-based groups, and each platform must allocate items to minimize total cost—defined by a convex combination of platform-level and group-level convex cost functions—while satisfying a lower bound on aggregate utility. This formulation flexibly balances overall efficiency and inter-group fairness. Crucially, convex cost functions replace rigid fairness constraints, unifying group fairness and Nash social welfare objectives within a single optimization framework. We propose a polynomial-time approximation algorithm based on linear programming and network flow techniques, with provable approximation guarantees. For instances with special structural properties, we design an exact algorithm; moreover, we prove NP-hardness for the general cross-group setting. Extensive experiments demonstrate the method’s effectiveness in trading off utility and fairness, as well as its scalability to large-scale instances.

Technology Category

• Game Theory and Economic Paradigms: Fair Division
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Combinatorial Optimization

Application Category

• Economics, Online Markets and Human Computation: Economics and fairness of platforms and recommendation systems
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

We consider the problem of assigning items to platforms where each item has a utility associated with each of the platforms to which it can be assigned. Each platform has a soft constraint over the total number of items it serves, modeled via a convex cost function. Additionally, items are partitioned into groups, and each platform also incurs group-specific convex cost over the number of items from each group that can be assigned to the platform. These costs promote group fairness by penalizing imbalances, yielding a soft variation of fairness notions introduced in prior work, such as Restricted Dominance and Minority protection. Restricted Dominance enforces upper bounds on group representation, while Minority protection enforces lower bounds. Our approach replaces such hard constraints with cost-based penalties, allowing more flexible trade-offs. Our model also captures Nash Social Welfare kind of objective. The cost of an assignment is the sum of the values of all the cost functions across all the groups and platforms. The objective is to find an assignment that minimizes the cost while achieving a total utility that is at least a user-specified threshold. The main challenge lies in balancing the overall platform cost with group-specific costs, both governed by convex functions, while meeting the utility constraint. We present an efficient polynomial-time approximation algorithm, supported by theoretical guarantees and experimental evaluation. Our algorithm is based on techniques involving linear programming and network flows. We also provide an exact algorithm for a special case with uniform utilities and establish the hardness of the general problem when the groups can intersect arbitrarily.

Problem

• Assign items to platforms with utility and fairness constraints
• Minimize convex cost functions for group and platform balance
• Develop efficient algorithms for fair matching with utility thresholds

Innovation

• Convex cost functions for group fairness
• Polynomial-time approximation algorithm
• Linear programming and network flows