Partial Optimality in the Preordering Problem

2026-02-19

AI Summary

This work addresses the challenge of efficiently identifying partial optimality in preorder optimization—specifically, detecting pairs of elements that cannot satisfy a given preorder relation in any optimal solution. We propose novel partial optimality conditions and develop an efficient verification algorithm grounded in combinatorial optimization and graph theory. The resulting method substantially enhances the ability to recognize non-preorderable element pairs, significantly increasing the proportion of such pairs that can be pruned efficiently on both real-world and synthetic datasets. This advancement provides a more powerful tool for preorder inference with direct applications in bioinformatics and social network analysis.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Knowledge Representation and Reasoning: Preferences

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

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.

Problem

• preordering
• partial optimality
• NP-hard
• clustering
• partial ordering

Innovation

• partial optimality
• preordering problem
• efficient algorithms
• NP-hard optimization
• bioinformatics