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Practical approach to $2$-Euclidean Preferences

Michal Dvořák, Dušan Knop, Jan Pokorný, Martin Slávik

TL;DR

This work develops a practical framework for recognizing whether a given election is $2$-Euclidean by combining a novel convex-hull forbidden substructure with reduction rules, ILP and QCP formulations, and targeted experiments on PrefLib. The approach yields substantial empirical gains, notably reducing unresolved PrefLib instances from $343$ to $60$ and solving $98.7\%$ of instances in under $1$ second. Key contributions include a convex-hull–based no-instance family, a refined embedding-graph perspective, and optimized ILP/QCP pipelines that outperform prior methods. The results demonstrate strong practical utility for identifying or refuting $2$-Euclidean preferences and offer a path toward higher-dimensional generalizations and further refinements. The work highlights substantial gaps between worst-case theory and real-world instances, showing that incomplete forbidden-structure characterizations can still enable efficient, scalable algorithms in practice, with broad implications for voting geometry and related decision problems.

Abstract

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 Tydrichová~[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.

Practical approach to $2$-Euclidean Preferences

TL;DR

This work develops a practical framework for recognizing whether a given election is -Euclidean by combining a novel convex-hull forbidden substructure with reduction rules, ILP and QCP formulations, and targeted experiments on PrefLib. The approach yields substantial empirical gains, notably reducing unresolved PrefLib instances from to and solving of instances in under second. Key contributions include a convex-hull–based no-instance family, a refined embedding-graph perspective, and optimized ILP/QCP pipelines that outperform prior methods. The results demonstrate strong practical utility for identifying or refuting -Euclidean preferences and offer a path toward higher-dimensional generalizations and further refinements. The work highlights substantial gaps between worst-case theory and real-world instances, showing that incomplete forbidden-structure characterizations can still enable efficient, scalable algorithms in practice, with broad implications for voting geometry and related decision problems.

Abstract

An election is a pair of candidates and voters. Each vote is a ranking (permutation) of the candidates. An election is -Euclidean if there is an embedding of both candidates and voters into such that voter prefers candidate over if and only if is closer to than is to in the embedding. For the problem of deciding whether is -Euclidean is -complete. In this paper, we propose practical approach to recognizing and refuting -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 Tydrichová~[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 to . Moreover, of PrefLib instances are resolved in under second using our approach.

Paper Structure

This paper contains 42 sections, 34 theorems, 31 equations, 17 figures.

Table of Contents

  1. Introduction
  2. Related Work
  3. Our contribution
  4. Paper Organization
  5. Preliminaries
  6. Geometry
  7. Graph theory
  8. Permutations
  9. Logic
  10. Elections
  11. Nice embeddings
  12. Forbidden substructure on $3$ voters ($3$-$8$ pattern)
  13. Convex Hull
  14. Reducing the number of candidates
  15. Removing candidates that behave similarly
  16. ...and 27 more sections

Key Result

lemma 1

Let $X$ be a finite set and $\gamma\colon X \to \mathbb{R}^2$ an embedding. Let $x_1,x_2\in X$ and $p_1,p_2\in\mathbb{R}^2$ be such that $x_1\neq x_2$ and $p_1\neq p_2$. Then there is a transformation $T$ preserving relative distances such that $(T\circ \gamma)(x_1)=p_1$ and $(T\circ \gamma)(x_2)=p_

Figures (17)

  • Figure 1: Examples of transformations. On the left, the solid triangle $\triangle a b c$ is mapped via a homothety with center $h$ and ratio $k=2$ to the dashed triangle $\triangle Ta Tb Tc$. We note that $h$ is chosen to be the circumcenter of the triangle $\triangle a b c$. Note that $h$ remains the circumcenter of the triangle $\triangle Ta Tb Tc$. This is because the circumcenter is the intersection of perpendicular bisectors of the three sides of a triangle. On the right the solid triangle $\triangle a b c$ is mapped via a rotation by angle $\theta=75^{\circ}$ to the dashed triangle $\triangle Ta Tb Tc$. Note that the center of rotation is always the origin.
  • Figure 2: The graph of permutations $G(\mathcal{S}_{C})$ for $C=\{a,b,c,d\}$ realised in the shape of a truncated octahedron. The edges of the graph correspond to consecutive swaps. Note that the distance $d^{\operatorname{swap}}$ coincides with the graph-theoretical distance $d_{G(\mathcal{S}_{C})}$. For example we have $d^{\operatorname{swap}}(abcd,dbca)=5$ because $dbca=abcd\circ \tau_{a,b} \circ \tau_{a,c}\circ \tau_{a,d}\circ \tau_{c,d}\circ \tau_{b,d}$ and $dbca$ cannot be expressed as composition of $abcd$ with fewer than $5$ consecutive swaps. These consecutive swaps correspond to one of the shortest paths from $abcd$ to $dbca$ in $G(\mathcal{S}_{C})$ given by $abcd,bacd,bcad,bcda,bdca,dbca$. The path is highlighted in red.
  • Figure 3: A $2$-Euclidean embedding of the election $(C,V)$ with $C=\{a,b,c,d\}$ and $V=\{bdac,bacd,adcb,acdb,dacb,cadb,bcad\}$. The dashes lines are the bisectors between pairs of candidates. The shaded triangle is the region $R^{\gamma}(adbc)$. Note that for example the region $R^{\gamma}(cdba)$ is empty.
  • Figure 4: The $3$-$8$ pattern.
  • Figure 5: The situation in the proof of \ref{['lem:two_controversial_consecutive_convex_hull']}.
  • ...and 12 more figures

Theorems & Definitions (39)

  • lemma 1
  • lemma 2
  • lemma 3
  • lemma 4
  • definition 1
  • theorem 1
  • lemma 5
  • definition 2
  • lemma 6
  • lemma 7
  • ...and 29 more