Optimal Metric Distortion for Matching on the Line

TL;DR

This paper introduces an algorithm for matching on the line that uses only ordinal preferences to achieve optimal or near-optimal social costs, improving understanding of metric distortion in matching problems.

Contribution

It presents a new algorithm for one-sided matching with provably optimal distortion and bounds on information needed for two-sided matching.

Findings

Achieves a distortion of 3 for one-sided matching, optimal for various social costs.

Shows that optimal two-sided matchings can be computed with minimal ordinal information.

Provides bounds on the number of queries needed for two-sided matching.

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.