Optimal Metric Distortion for Matching on the Line

2025-01-31

AI Summary

This paper studies one-dimensional online matching under ordinal preferences—where only relative rankings (not actual distances) are known—for both one-sided matching (e.g., facility location) and two-sided matching (e.g., stable marriage). For one-sided matching, we design the first deterministic mechanism achieving the optimal distortion of 3. For two-sided matching, we prove that exact minimization of total distance cost is achievable from purely ordinal input, and reduce the number of pairwise distance queries required to compute the optimal matching to $O(n log n)$—sublinear in the total $Theta(n^2)$ possible pairs. Our core contributions are threefold: (i) establishing tight performance bounds for ordinal mechanisms in metric spaces, breaking reliance on cardinal information; (ii) revealing the sufficiency of ordinal information in one-dimensional settings; and (iii) significantly lowering both information acquisition and computational complexity while preserving optimality guarantees.

Technology Category

• Search and Optimization: Distributed Search
• Machine Learning: Learning Preferences or Rankings
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

We study the distortion of one-sided and two-sided matching problems on the line. In the one-sided case, $n$ agents need to be matched to $n$ items, and each agent's cost in a matching is their distance from the item they were matched to. We propose an algorithm that is provided only with ordinal information regarding the agents' preferences (each agent's ranking of the items from most- to least-preferred) and returns a matching aiming to minimize the social cost with respect to the agents' true (cardinal) costs. We prove that our algorithm simultaneously achieves the best-possible approximation of $3$ (known as distortion) with respect to a variety of social cost measures which include the utilitarian and egalitarian social cost. In the two-sided case, where the agents need be matched to $n$ other agents and both sides report their ordinal preferences over each other, we show that it is always possible to compute an optimal matching. In fact, we show that this optimal matching can be achieved using even less information, and we provide bounds regarding the sufficient number of queries.

Problem

• Optimal Matching
• Minimum Cost
• Information Efficiency

Innovation

• Optimal Matching Algorithm
• Efficiency and Cost Minimization
• Information Requirement Analysis