In this paper, we construct and compare algorithmic approaches to solve the Preference Consistency Problem for preference statements based on hierarchical models. Instances of this problem contain a set of preference statements that are direct comparisons (strict and non-strict) between some alternatives, and a set of evaluation functions by which all alternatives can be rated. An instance is consistent based on hierarchical preference models, if there exists an hierarchical model on the evaluation functions that induces an order relation on the alternatives by which all relations given by the preference statements are satisfied. Deciding if an instance is consistent is known to be NP-complete for hierarchical models. We develop three approaches to solve this decision problem. The first involves a Mixed Integer Linear Programming (MILP) formulation, the other two are recursive algorithms that are based on properties of the problem by which the search space can be pruned. Our experiments on synthetic data show that the recursive algorithms are faster than solving the MILP formulation and that the ratio between the running times increases extremely quickly.

翻译：本文针对基于层次模型的偏好陈述，构建并比较了解决偏好一致性问题的算法方法。该问题的实例包含一组偏好陈述（即某些备选方案之间的严格与非严格直接比较）以及一组能够对所有备选方案进行评分的评价函数。若存在一个基于评价函数的层次模型，该模型能够导出一个满足所有偏好陈述所给关系的备选方案序关系，则称该实例在层次偏好模型下是一致的。已知对于层次模型，判定实例一致性是NP完全问题。我们提出了三种解决该判定问题的方法：第一种采用混合整数线性规划（MILP）建模；另外两种是基于问题特性的递归算法，可通过剪枝缩减搜索空间。在合成数据上的实验表明，递归算法比求解MILP模型更快，且两者运行时间的比值呈极快速增长趋势。