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

The Kemeny aggregation problem consists of computing the consensus rankings of an election with respect to the Kemeny-Young voting method. These aggregated rankings are the geometric medians as well as the maximum likelihood estimators in the Mallows model of the rankings in the election under the Kendall-tau distance which counts the number of pairwise disagreements. The problem admits fundamental applications in various domains such as computational social choice, machine learning, operations research, and biology but its computational complexity is unfortunately expensive. In this paper, we establish optimized quantitative extensions of the well-known 3/4-majority rule of Betzler et al. and the Major Order Theorem of Hamel and Milosz for the Kemeny aggregation problem. By taking into account the extra information available in the problem such as the number of candidates and by considering an additional optimization of certain piecewise linear functions in one variable, our results achieve significantly more refined space reduction techniques as illustrated by experimental results on real and synthetic data without increasing the time complexity of the algorithms.