Recognizing when a preference system is close to admitting a master list

TL;DR

This paper investigates the complexity of determining whether a preference system can be made to admit a master list through limited modifications, with implications for stable and popular matchings.

Contribution

It introduces the first complexity analysis of how close preference systems are to admitting a master list via various modifications.

Findings

Decides the complexity of preference system modifications for admitting a master list.

Analyzes parameterized complexity and approximation for these modification problems.

Provides applications to stable and popular matchings.

Abstract

A preference system $\mathcal{I}$ is an undirected graph where vertices have preferences over their neighbors, and $\mathcal{I}$ admits a master list if all preferences can be derived from a single ordering over all vertices. We study the problem of deciding whether a given preference system $\mathcal{I}$ is close to admitting a master list based on three different distance measures. We determine the computational complexity of the following questions: can $\mathcal{I}$ be modified by (i) $k$ swaps in the preferences, (ii) $k$ edge deletions, or (iii) $k$ vertex deletions so that the resulting instance admits a master list? We investigate these problems in detail from the viewpoint of parameterized complexity and of approximation. We also present two applications related to stable and popular matchings.