Recognizing 1-Euclidean Preferences: An Alternative Approach
We consider the problem of detecting whether a given election is 1-Euclidean, i.e., whether voters and candidates can be mapped to points on the real line so that voters’ preferences over the candidates are determined by the Euclidean distance. A recent paper by Knoblauch  shows that this problem admits a polynomial-time algorithm. Knoblauch’s approach relies on the fact that a 1-Euclidean election is necessarily single-peaked, and makes use of the properties of the respective candidate order to find a mapping of voters and candidates to the real line. We propose an alternative polynomial-time algorithm for this problem, which is based on the observation that a 1-Euclidean election is necessarily singe-crossing, and we use the properties of the respective voter order to find the desired mapping.
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