Space reduction techniques for the -wise Kemeny problem

Kemeny's rule is one of the most studied and well-known voting schemes with various important applications in computational social choice and biology. Recently, Kemeny's rule was generalized via a set-wise approach by Gilbert et. al. Following this paradigm, we have shown in \cite{Phung-Hamel-2023} that the -wise Kemeny voting scheme induced by the -wise Kendall-tau distance presents interesting advantages in comparison with the classical Kemeny rule. While the -wise Kemeny problem, which consists of computing the set of -wise consensus rankings of a voting profile, is NP-hard, we establish in this paper several generalizations of the Major Order Theorems, as obtained in \cite{Milosz-Hamel-2020} for the classical Kemeny rule, for the -wise Kemeny voting scheme to achieve a substantial search space reduction by efficiently determining in polynomial time the relative orders of pairs of alternatives. Essentially, our theorems quantify precisely the non-trivial property that if the preference for an alternative over another one in an election is strong enough, not only in the head-to-head competition but even when taking into consideration one or two more alternatives, then the relative order of these two alternatives in every -wise consensus ranking must be as expected. Moreover, we show that the well-known -majority rule of Betzler et al. for the classical Kemeny rule is only valid for elections with no more than alternatives with respect to the -wise Kemeny scheme. Examples are also provided to show that the -wise Kemeny rule is more resistant to manipulation than the classical one.
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