Partial Optimality in the Preordering Problem

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.