Efficient space reduction techniques by optimized majority rules for the Kemeny aggregation problem and beyond

The Kemeny aggregation problem consists of computing the consensus rankings of an election with respect to the well-known Kemeny-Young voting method. These consensus rankings satisfy various fundamental properties and are the geometric medians of the votes in the election under the Kendall-tau distance which counts the number of pairwise disagreements. The Kemeny aggregation problem admits important applications in various domains such as computational social choice, machine learning, operations research, and biology but it is unfortunately NP-hard. Recently, Milosz and the second author presented an approach to reduce the search space of the problem by solving the relative order of pairs of elements in those consensus. In this article, we prove an optimized extension of this approach achieving significantly more refined space reduction techniques without adding much to the running time of the algorithms in practice, as illustrated by experimental results and analysis on real and synthetic data. We show how the constraints built by our approach can be used in combination with other methods such as Integer Programming and Finest Condorcet Partitioning to achieve an efficient and scalable solution approach to the Kemeny aggregation problem. Relaxed and approximate versions of our algorithms are also described and evaluated. We also provide practical methods to compute provable guarantees for the quality of the approximate rankings obtained.