Partial Optimality in the Preordering Problem

TL;DR

This paper introduces new partial optimality conditions and algorithms for the NP-hard preordering problem, improving the efficiency of identifying non-ordered pairs in real-world and synthetic datasets.

Contribution

It provides novel partial optimality conditions and efficient algorithms that enhance the solution process for the preordering problem.

Findings

New conditions increase the fraction of pairs efficiently identified as not in the preorder.

Algorithms improve partial optimality detection in real and synthetic data.

Enhanced partial solutions contribute to solving the NP-hard preordering problem.

Abstract

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.