Diversity of Structured Domains via k-Kemeny Scores

In the k-KEMENY problem, we are given an ordinal election, i.e., a collection of votes ranking the candidates from best to worst, and we seek the smallest number of swaps of adjacent candidates that ensure that the election has at most k different rankings. We study this problem for a number of structured domains, including the single-peaked, single-crossing, group-separable, and Euclidean ones. We obtain two kinds of results: (1) We show that k-KEMENY remains intractable under most of these domains, even for k = 2, and (2) we use k-KEMENY to rank these domains in terms of their diversity.