Practical approach to $2$-Euclidean Preferences

An election is a pair $(C,V)$ of candidates and voters. Each vote is a ranking (permutation) of the candidates. An election is $d$-Euclidean if there is an embedding of both candidates and voters into $\mathbb{R}^d$ such that voter $v$ prefers candidate $a$ over $b$ if and only if $a$ is closer to $v$ than $b$ is to $v$ in the embedding. For $d\geq 2$ the problem of deciding whether $(C,V)$ is $d$-Euclidean is $\exists \mathbb{R}$-complete. In this paper, we propose practical approach to recognizing and refuting $2$-Euclidean preferences. We design a new class of forbidden substructures that works very well on practical instances. We utilize the framework of integer linear programming (ILP) and quadratically constrained programming (QCP). We also introduce reduction rules that simplify many real-world instances significantly. Our approach beats the previous algorithm of Escoffier, Spanjaard and Tydrichov\'a~[Algorithmic Recognition of 2-Euclidean Preferences, ECAI 2023] both in number of resolved instances and the running time. In particular, we were able to lower the number of unresolved PrefLib instances from $343$ to $60$. Moreover, $98.7\%$ of PrefLib instances are resolved in under $1$ second using our approach.