Partial Optimality in the Preordering Problem
This work provides incremental improvements for researchers in bioinformatics and social network analysis dealing with preordering, a generalization of clustering and partial ordering.
The paper tackles the NP-hard preordering problem by developing new partial optimality conditions and efficient algorithms to decide them, resulting in an increased fraction of pairs for which it is efficiently determined that one element does not precede another in an optimal preorder in experiments with real and synthetic data.
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.