Weighted majority tournaments and Kemeny ranking with 2-dimensional Euclidean preferences

The assumption that voters' preferences share some common structure is a standard way to circumvent NP-hardness results in social choice problems. While the Kemeny ranking problem is NP-hard in the general case, it is known to become easy if the preferences are 1-dimensional Euclidean. In this note, we prove that the Kemeny ranking problem remains NP-hard for $k$-dimensional Euclidean preferences with $k\!\ge\!2$ under norms $\ell_1$, $\ell_2$ and $\ell_\infty$, by showing that any weighted tournament (resp. weighted bipartite tournament) with weights of same parity (resp. even weights) is inducible as the weighted majority tournament of a profile of 2-Euclidean preferences under norm $\ell_2$ (resp. $\ell_1,\ell_{\infty}$), computable in polynomial time. More generally, this result regarding weighted tournaments implies, essentially, that hardness results relying on the (weighted) majority tournament that hold in the general case (e.g., NP-hardness of Slater ranking) are still true for 2-dimensional Euclidean preferences.

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