Reliability of Solutions in Linear Ordering Problem: New Probabilistic Insight and Algorithms

In this work, our goal is to characterize the reliability of the solutions that can be obtained by the linear ordering problem (LOP) which is used to order $M$ objects from their pairwise comparisons. We adopt a probabilistic perspective, where the results of pairwise comparisons are modeled as Bernoulli variables with a common parameter which we estimate from the observed data. Estimation by brute-force enumeration has a prohibitive complexity of O($M!$) we thus reformulate the problem and introduce a concept of Slater's spectrum which generalizes Slater's index, and next, devising an efficient algorithm to find the spectrum, we lower the complexity to O($M^2 2^M$) which is manageable for moderate-size LOPs. Furthermore, with a minor modification of the algorithm, we are able to find all solutions of the LOP. Numerical examples on synthetic and real-world data are shown and the Python-implemented algorithms are publicly available.