# Computing Optimal Assignments in Linear Time for Approximate Graph Matching

## TL;DR

This paper introduces a linear-time algorithm for optimal assignment problems when costs are tree-based, enabling efficient approximation of graph edit distances in large datasets.

## Contribution

The paper presents a novel linear-time algorithm for optimal assignment with tree-based costs, improving efficiency in graph matching tasks.

## Key findings

- Algorithm finds optimal assignment in linear time for tree-based costs.
- Effective approximation of graph edit distance demonstrated on real-world data.
- Method handles both structural and continuous vertex label differences.

## Abstract

Finding an optimal assignment between two sets of objects is a fundamental problem arising in many applications, including the matching of `bag-of-words' representations in natural language processing and computer vision. Solving the assignment problem typically requires cubic time and its pairwise computation is expensive on large datasets. In this paper, we develop an algorithm which can find an optimal assignment in linear time when the cost function between objects is represented by a tree distance. We employ the method to approximate the edit distance between two graphs by matching their vertices in linear time. To this end, we propose two tree distances, the first of which reflects discrete and structural differences between vertices, and the second of which can be used to compare continuous labels. We verify the effectiveness and efficiency of our methods using synthetic and real-world datasets.