A survey on the complexity of tournament solutions

In voting theory, the result of a paired comparison method such as the one suggested by Condorcet can be represented by a tournament, i.e., a complete asymmetric directed graph. When there is no Condorcet winner, i.e., a candidate preferred to any other candidate by a majority of voters, it is not always easy to decide who is the winner of the election. Different methods, called tournament solutions, have been proposed for defining the winners. They differ in their properties and usually lead to different winners. Among these properties, we consider in this survey the algorithmic complexity of the most usual tournament solutions: some are polynomial, some are NP-hard, while the complexity status of others remains unknown.