Complexity of computing median linear orders and variants

International audience<p>Given a finite set X and a collection C of linear orders defined on X, computinga median linear order (Condorcet-Kemeny's problem) consists in determining alinear order O minimizing the remoteness from C. This remoteness is based on thesymmetric distance, and measures the number of disagreements between O and C.In the context of voting theory, X can be considered as a set of candidates andthe linear orders of C as the preferences of voters, while a linear order minimizing the remoteness from C can be adopted as the collective ranking of the candidates with respect to the voters' opinions. This paper studies the complexity of this problem and of several variants of it: computing a median order, computing a winner according to this method, checking that a given candidate is a winner and so on. We try to locate these problems inside the polynomial hierarchy.