  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. This paradigm presents interesting advantages in comparison with Kemeny's
rule since not only pairwise comparisons but also the discordance between the
winners of subsets of three alternatives are also taken into account in the
definition of the $3$-wise Kendall-tau distance between two rankings. In spite
of the NP-hardness of the 3-wise Kemeny problem which consists of computing the
set of $3$-wise consensus rankings, namely rankings whose total $3$-wise
Kendall-tau distance to a given voting profile is minimized, we establish in
this paper several generalizations of the Major Order Theorems, as obtained by
Milosz and Hamel for Kemeny's rule, for the $3$-wise Kemeny voting schemes 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 nontrivial 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 account one or two more
alternatives, then the relative order of these two alternatives in all $3$-wise
consensus rankings must be as expected. As an application, we also obtain an
improvement of the Major Order Theorems for Kememy's rule. Moreover, we show
that the well-known $3/4$-majority rule of Betzler et al. for Kemeny's rule is
only valid in general for elections with no more than $5$ alternatives with
respect to the $3$-wise Kemeny scheme. Several simulations and tests of our
algorithms on real-world and uniform data are provided.
