面向基于层次模型的偏好一致性问题的快速算法研究
Towards Fast Algorithms for the Preference Consistency Problem Based on Hierarchical Models
AI总结：
针对层次模型下NP完全的偏好一致性判定问题，提出MILP公式法和两种基于搜索空间剪枝的递归算法，实验证明递归算法速度更快且运行时间比值增长极快。
摘要：
本文构建并对比了用于求解基于层次模型的偏好陈述的偏好一致性问题的多种算法方案。该问题的实例包含一组偏好陈述（即若干备选方案之间的严格与非严格直接比较），以及一组可对所有备选方案进行评分的评估函数。若评估函数上存在一个层次模型，其能在备选方案上诱导出一种序关系，且满足偏好陈述给出的所有关系，则该实例在层次偏好模型下是一致的。已知判定实例是否一致的问题在层次模型下属于NP完全问题。我们提出了三种求解该判定问题的方法：第一种采用混合整数线性规划（MILP，Mixed Integer Linear Programming）公式，另外两种是递归算法，基于可缩减搜索空间的问题特性设计。我们在合成数据上开展的实验表明，递归算法的求解速度快于MILP公式法，且两者的运行时间比值增长极快。
英文摘要：
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.
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