Two algorithms for finding optimal solutions of the Kemeny rank aggregation problem for full rankings

Abstract

The analysis of ranking data has recently received increasing attention in many fields (i.e. political sciences, computer sciences, social sciences, medical sciences, etc.).Typically when dealing with preference rankings one of the main issue is to find a ranking that best represents the set of input rankings.Among several measures of agreement proposed in the literature, the Kendall's distance is probably the most known. We propose a branch-and-bound algorithm to find the solution(s) even when we take into account a relatively large number of objects to be ranked. We also propose a heuristic variant of the branch-and-bound algorithm useful when the number of objects to rank is particularly high. We show how the solution(s) achieved by the algorithm can be employed in different analysis of rank data such as Mallow's phi model, mixtures of distance-based models, cluster analysis and so on.