Preordering is a generalization of clustering and partial ordering with applications in bioinformatics and social network analysis. Given a finite set $V$ and a value $c_{ab} \in \mathbb{R}$ for every ordered pair $ab$ of elements of $V$, the preordering problem asks for a preorder $\lesssim$ on $V$ that maximizes the sum of the values of those pairs $ab$ for which $a \lesssim b$. Building on the state of the art in solving this NP-hard problem partially, we contribute new partial optimality conditions and efficient algorithms for deciding these conditions. In experiments with real and synthetic data, these new conditions increase, in particular, the fraction of pairs $ab$ for which it is decided efficiently that $a \not\lesssim b$ in an optimal preorder.

翻译：预排序是聚类与偏序的推广，在生物信息学和社会网络分析中具有应用价值。给定有限集合$V$及其元素间每个有序对$ab$对应的值$c_{ab} \in \mathbb{R}$，预排序问题要求找到$V$上的一个预序关系$\lesssim$，使得满足$a \lesssim b$的配对$ab$的值之和达到最大。基于当前解决这一NP难问题的部分最优性技术，我们提出了新的部分最优性条件，并设计了判定这些条件的高效算法。在真实数据与合成数据的实验中，这些新条件显著提升了可高效判定“在最优预序中$a \not\lesssim b$”的配对$ab$的比例。