Necessarily Optimal One-Sided Matchings

TL;DR

This paper investigates algorithms for determining necessarily Pareto optimal and rank-maximal matchings from partial top-$k$ preferences, providing efficient checks and online elicitation strategies with competitive bounds.

Contribution

It introduces methods to verify and find necessarily optimal matchings from partial preferences and develops online algorithms with provable performance guarantees.

Findings

Efficient algorithms for checking NPO and NRM matchings.

Existence checks for NPO and NRM matchings under partial preferences.

Online algorithms with competitive ratio bounds for preference elicitation.

Abstract

We study the classical problem of matching $n$ agents to $n$ objects, where the agents have ranked preferences over the objects. We focus on two popular desiderata from the matching literature: Pareto optimality and rank-maximality. Instead of asking the agents to report their complete preferences, our goal is to learn a desirable matching from partial preferences, specifically a matching that is necessarily Pareto optimal (NPO) or necessarily rank-maximal (NRM) under any completion of the partial preferences. We focus on the top-$k$ model in which agents reveal a prefix of their preference rankings. We design efficient algorithms to check if a given matching is NPO or NRM, and to check whether such a matching exists given top-$k$ partial preferences. We also study online algorithms for eliciting partial preferences adaptively, and prove bounds on their competitive ratio.