Rank aggregation is widely used in group decision-making and many other applications where it is of interest to consolidate heterogeneous ordered lists. Oftentimes, these rankings may involve a large number of alternatives, contain ties, and/or be incomplete, all of which complicate the use of robust aggregation methods. In particular, these characteristics have limited the applicability of the aggregation framework based on the Kemeny-Snell distance, which satisfies key social choice properties that have been shown to engender improved decisions. This work introduces a binary programming formulation for the generalized Kemeny rank aggregation problem---whose ranking inputs may be complete and incomplete, with and without ties---and compare it to a modified version of a recently developed integer programming formulation for the generalized Kendall-tau distance. The new formulation has fewer variables and constraints, which leads to faster solution times. Moreover, we develop a new social choice property, the Non-strict Extended Condorcet Criterion, which can be regarded as a natural extension of the well-known Condorcet criterion and the Extended Condorcet criterion. Unlike its parent properties, the new property is adequate for handling complete rankings with ties. The property is leveraged to develop a structural decomposition algorithm, through which certain large instances of the NP-hard Kemeny rank aggregation problem can be solved exactly in a practical amount of time. To test the practical implications of the new formulation and social choice property, we work with instances constructed from a probabilistic distribution and benchmark instances from PrefLib, a library of preference data.
Yoo, Y., & Escobedo, A. R. (2020). A new binary programming formulation and social choice property for Kemeny rank aggregation.