Rankability and Linear Ordering Problem: New Probabilistic Insight and Algorithms

TL;DR

This paper introduces a probabilistic approach to assess rankability in the linear ordering problem, proposing new algorithms based on the Slater spectrum that efficiently handle moderate-sized data sets.

Contribution

It develops a probabilistic framework for the linear ordering problem and introduces algorithms to compute the Slater spectrum with manageable complexity.

Findings

The algorithms effectively determine rankability in synthetic and real data.

The Slater spectrum generalizes the Slater index for better data interpretation.

Proposed methods are computationally feasible for moderate M.

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 are rankable, that is, if the linear ordering problem (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 brute-force approach to the required enumeration has a prohibitive complexity of O(M !), so we 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(M^3 2^M) 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(M 2^M). Numerical examples are shown on synthetic and real-world data, and the algorithms are publicly available.