Solving the Kemeny ranking aggregation problem with quantum optimization algorithms

The aim of a ranking aggregation problem is to combine several rankings into a single one that best represents them. A common method for solving this problem is due to Kemeny and selects as the aggregated ranking the one that minimizes the sum of the Kendall distances to the rankings to be aggregated. Unfortunately, the identification of the said ranking-called the Kemeny ranking-is known to be a computationally expensive task. In this paper, we study different ways of computing the Kemeny ranking with quantum optimization algorithms, and in particular, we provide some alternative formulations for the search for the Kemeny ranking as an optimization problem. To the best of our knowledge, this is the first time that this problem is addressed with quantum techniques. We propose four different ways of formulating the problem, one novel to this work. Two different quantum optimization algorithms-Quantum Approximate Optimization Algorithm and Quantum Adiabatic Computing-are used to evaluate each of the different formulations. The experimental results show that the choice of the formulation plays a big role on the performance of the quantum optimization algorithms.