Median of an odd number of permutations

Given m permutations π1, π2 . . . πm of {1, 2, . . . , n} and a distance function d, the median problem is to find a permutation π ∗ that is the "closest" of the m given permutations. Here, we study the problem under the Kendall-τ distance that counts the number of pairwise disagreements between permutations. This problem is also known, in the context of rank aggregation, as the Kemeny Score Problem and has been proved to be NP-hard when m ≥ 4. This article is an extension of [4], where some nice combinatorial properties of the case m = 3 where stated without proof, to the general case m ≥ 3, m odd.