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.

翻译：线性排序问题（Linear Ordering Problem, LOP）旨在通过对象间的成对比较结果对M个对象进行排序，广泛应用于多个研究领域。尽管已开发出高效的LOP算法，但关于数据是否具有可排序性（即LOP解是否具有实际解释意义）的验证工作却鲜有研究。针对这一问题，我们采用概率视角，将成对比较结果建模为具有共同参数的伯努利变量，并通过观测数据估计该参数。直接枚举的暴力方法计算复杂度高达O(M!)，因此我们重新构建问题，引入广义斯莱特指数的斯莱特谱（Slater spectrum）概念，进而设计出复杂度为O(M^3 2^M)的算法，该算法可在中等M值下实现可行计算。此外，通过对算法进行微小修改，我们能够以O(M 2^M)的复杂度找到LOP的所有解。基于合成数据与实际数据的数值示例验证了算法的有效性，且相关算法已公开提供。