Fairness and the set of optimal rankings for the linear ordering problem

The goals of this paper are: (1) to bring attention to the existence and utility of multiple optimal rankings for the linear ordering problem, (2) to make the case for finding some or all of these multiple optimal rankings, (3) to provide an efficient algorithm that determines the existence of multiple optimal rankings, (4) to provide algorithms that find a sample of all optimal rankings, and (5) to connect multiple optimal rankings to fairness in ranking. We create algorithms to find the two nearest optimal rankings, the two farthest optimal rankings, and a so-called centroid ranking nearest to the centroid, which summarizes the information in all optimal rankings.