On Maximum Rank Aggregation Problems

The rank aggregation problem consists in finding a consensus ranking on a set of alternatives, based on the preferences of individual voters. These are expressed by permutations, whose distance can be measured in many ways. In this work we study a collection of distances, including the Kendall tau, Spearman footrule, Spearman rho, Cayley, Hamming, Ulam, and Minkowski distances, and compute the consensus against the maximum, which attempts to minimize the discrimination against any voter. We provide a general schema from which we can derive the NP-hardness of the maximum rank aggregation problems under the aforementioned distances. This reveals a dichotomy for rank aggregation problems under the Spearman footrule and Minkowski distances: the common sum version is solvable in polynomial time whereas the maximum version is NPhard. Moreover, the maximum rank aggregation problems are proved to be 2-approximable under all pseudometrics and fixed-parameter tractable under the Kendall tau, Hamming, and Minkowski distances.