Rankability and linear ordering problem: Probabilistic insight and algorithms

Abstract

The linear ordering problem (LOP), which consists in ordering M objects from their pairwise comparisons, is commonly applied in many areas of research. While efforts have been made to devise efficient LOP algorithms, verification of whether the data is rankable, that is, if the LOP solutions have a meaningful interpretation, received much less attention. To address this problem, we adopt a probabilistic perspective where the results of pairwise comparisons are modeled as Bernoulli variables with a common parameter, and we estimate the latter from the observed data. The enumeration over the space of permutations, required to solve the problem, has a prohibitive complexity of O(M!) if implemented explicitly. We thus reformulate the problem and introduce a concept of the Slater spectrum that generalizes the Slater index, and then devise an algorithm to find the spectrum with complexity O(M32M) that is manageable for moderate values of M. Furthermore, with a minor modification of the algorithm, we are able to find all solutions of the LOP with the complexity O(M2M). Numerical examples are shown on synthetic and real-world data, and the algorithms are publicly available.