AAAI Publications, Thirty-First AAAI Conference on Artificial Intelligence

Font Size: 
Recognising Multidimensional Euclidean Preferences
Dominik Peters

Last modified: 2017-02-10


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 preferences and the behaviour of other metrics, and survey a variety of related work.


social choice; voting; single-peaked preferences; spatial preferences; recognition problem; computational complexity, ETR; forbidden subprofiles; multidimensional unfolding

Full Text: PDF