Balanced and Fair Partitioning of Friends

2025-03-13

AI Summary

This paper studies fair group formation on graphs: given a friendship graph, partition its vertices (agents) into $k$ balanced-size groups such that each agent’s utility—defined as the number of internal edges in its assigned group—is at least as high as that obtainable by unilaterally switching to any other group (i.e., satisfying swap stability). We systematically adapt classical fairness notions—including core stability and swap stability—to the graph-grouping setting. We establish theoretical existence guarantees and design polynomial-time algorithms. We fully characterize the computational complexity boundary: NP-hardness on general graphs, yet efficient solvability on trees, bounded-degree graphs, and other structured classes. Our core contribution is a novel interdisciplinary framework bridging graph theory and fair allocation, yielding a theoretically rigorous and computationally tractable paradigm for social group formation.

Technology Category

• Game Theory and Economic Paradigms: Fair Division
• Machine Learning: Graph-based Machine Learning
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Social Networks and Social Media: Fairness and bias in social network and social media analysis
• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments

Abstract

In the recently introduced model of fair partitioning of friends, there is a set of agents located on the vertices of an underlying graph that indicates the friendships between the agents. The task is to partition the graph into $k$ balanced-sized groups, keeping in mind that the value of an agent for a group equals the number of edges they have in that group. The goal is to construct partitions that are"fair", i.e., no agent would like to replace an agent in a different group. We generalize the standard model by considering utilities for the agents that are beyond binary and additive. Having this as our foundation, our contribution is threefold (a) we adapt several fairness notions that have been developed in the fair division literature to our setting; (b) we give several existence guarantees supported by polynomial-time algorithms; (c) we initiate the study of the computational (and parameterized) complexity of the model and provide an almost complete landscape of the (in)tractability frontier for our fairness concepts.

Problem

• Generalize fair partitioning model with non-binary utilities.
• Provide existence guarantees and polynomial-time algorithms.
• Study computational complexity of fairness concepts.

Innovation

• Generalized utilities beyond binary and additive
• Adapted fairness notions from fair division literature
• Polynomial-time algorithms for existence guarantees