On the hardness of 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. The alternatives are expressed by permutations, whose pairwise distance can be measured in many ways. In this work we study a collection of distances, including the Kendall tau, Spearman footrule, Minkowski, Cayley, Hamming, Ulam, and related edit distances. Unlike the common median by summation, we compute the consensus against the maximum. The maximum consensus attempts to minimize the discrimination against any voter and is a smallest enclosing ball or center problem. We provide a general schema via local permutations for the NP-hardness of the maximum rank aggregation problems under all distances which satisfy some general requirements. This unifies former NP-hardness results for some distances and lays the ground for further ones. In particular, we establish a dichotomy for rank aggregation problems under the Spearman footrule and Minkowski distances: The median version is solvable in polynomial time whereas the maximum version is NP-hard. Moreover, we show that the maximum rank aggregation problem is 2-approximable under any pseudometric and fixed-parameter tractable under the Kendall tau, Hamming, and Minkowski distances, where again a general schema via modification sets applies.