Recognising Multidimensional Euclidean Preferences

TL;DR

This paper investigates the computational complexity of recognizing multidimensional Euclidean preferences, showing NP-hardness for dimensions greater than one and exploring related theoretical properties and special cases.

Contribution

It proves that recognizing d-Euclidean preferences for d > 1 is NP-hard and establishes several theoretical results about the complexity and structure of these preferences.

Findings

Recognition problem is NP-hard for d > 1

Some profiles require exponentially many bits for embedding

No finite forbidden minor characterization exists for d > 1

Abstract

Euclidean preferences are a widely studied preference model, in which decision makers and alternatives are embedded in d-dimensional Euclidean space. Decision makers prefer those alternatives closer to them. This model, also known as multidimensional unfolding, has applications in economics, psychometrics, marketing, and many other fields. We study the problem of deciding whether a given preference profile is d-Euclidean. For the one-dimensional case, polynomial-time algorithms are known. We show that, in contrast, for every other fixed dimension d > 1, the recognition problem is equivalent to the existential theory of the reals (ETR), and so in particular NP-hard. We further show that some Euclidean preference profiles require exponentially many bits in order to specify any Euclidean embedding, and prove that the domain of d-Euclidean preferences does not admit a finite forbidden minor characterisation for any d > 1. We also study dichotomous preferencesand the behaviour of other metrics, and survey a variety of related work.