最大序数二因子分解
Maximal Ordinal Two-Factorizations
AI总结：
本研究探讨形式背景中的序数二因子分解，证明判定给定大小二因子分解的存在性是NP完全问题，并提出算法Ord2Factor以高效计算大型序数二因子分解。
摘要：
给定一个形式背景，序数因子是其关联关系的一个子集，该子集在概念格中形成一条链，即数据集中对应于线性顺序的部分。为了可视化形式背景中的数据，Ganter和Glodeanu提出了一种基于两个序数因子的双标图。为了使该双标图有用，重要的是这些因子尽可能多地包含数据点，即它们覆盖关联关系的很大一部分。在这项工作中，我们研究了此类序数二因子分解。首先，我们针对省略序数二因子分解的形式背景，研究了两个因子的不相交性。然后，我们证明了判定给定大小的二因子分解是否存在是一个NP完全问题，这使得计算最大因子分解在计算上代价高昂。最后，我们提供了算法Ord2Factor，该算法允许我们计算大型序数二因子分解。
英文摘要：
Given a formal context, an ordinal factor is a subset of its incidence relation that forms a chain in the concept lattice, i.e., a part of the dataset that corresponds to a linear order. To visualize the data in a formal context, Ganter and Glodeanu proposed a biplot based on two ordinal factors. For the biplot to be useful, it is important that these factors comprise as much data points as possible, i.e., that they cover a large part of the incidence relation. In this work, we investigate such ordinal two-factorizations. First, we investigate for formal contexts that omit ordinal two-factorizations the disjointness of the two factors. Then, we show that deciding on the existence of two-factorizations of a given size is an NP-complete problem which makes computing maximal factorizations computationally expensive. Finally, we provide the algorithm Ord2Factor that allows us to compute large ordinal two-factorizations.
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