Robust Popular Matchings

2024-01-23

Adaptive Agents and Multi-Agent Systems

AI Summary

This paper investigates robust popularity in preference matching: given multiple slightly perturbed preference instances, does there exist a matching that is popular across all instances? We consider two perturbation models: (1) minor adjustments to a single agent’s preference list, and (2) unavailability of certain alternatives (incomplete preferences). We introduce the notion of *robust popular matching* and establish a fine-grained complexity dichotomy based on perturbation structure. Specifically, we show that existence is polynomial-time decidable under arbitrary single-agent perturbations, yet becomes NP-complete even when only four agents each shift one alternative downward by one position. For the alternative-unavailability model, we provide a precise P/NP dichotomy. Our approach integrates combinatorial matching theory, graph algorithm design, and computational complexity analysis, identifying critical thresholds governing robustness.

Technology Category

• Knowledge Representation and Reasoning: Preferences
• Machine Learning: Learning Preferences or Rankings
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems
• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments

Abstract

We study popularity for matchings under preferences. This solution concept captures matchings that do not lose against any other matching in a majority vote by the agents. A popular matching is said to be robust if it is popular among multiple instances. We present a polynomial-time algorithm for deciding whether there exists a robust popular matching if instances only differ with respect to the preferences of a single agent while obtaining NP-completeness if two instances differ only by a downward shift of one alternative by four agents. Moreover, we find a complexity dichotomy based on preference completeness for the case where instances differ by making some options unavailable.

Problem

• Finding robust popular matchings across multiple preference instances
• Analyzing computational complexity of robust matching variations
• Developing algorithms for stable matching under preference changes

Innovation

• Polynomial-time algorithm for robust popular matchings
• Handles preference changes for single agent instances
• NP-completeness for multi-agent preference alterations