Space reduction techniques for the 3-wise Kemeny problem
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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 <cit.> that the 3-wise Kemeny voting scheme induced by the 3-wise Kendall-tau distance presents interesting advantages in comparison with the classical Kemeny rule. While the 3-wise Kemeny problem, which consists of computing the set of 3-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 <cit.> for the classical Kemeny rule, for the 3-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 3-wise consensus ranking must be as expected. Moreover, we show that the well-known 3/4-majority rule of Betzler et al. for the classical Kemeny rule is only valid for elections with no more than 5 alternatives with respect to the 3-wise Kemeny scheme. Examples are also provided to show that the 3-wise Kemeny rule is more resistant to manipulation than the classical one.
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