Symmetry and Similarity

Introduction

The goal of this project is to develop an algorithmic theory of similarity between graphs. Graphs are versatile models for representing complex data ranging from chemical molecules to social interactions. Dealing with graphical data and enabling modern data analysis techniques, a fundamental task is to compare graphs and to measure their similarity, preferably in a semantically meaningful and algorithmically efficient way.

Problem Statement

However, it is not clear at all how to achieve this. In many application areas, for example, computer vision, database systems, and formal verification, researchers have proposed (often ad-hoc) solutions to this problem tailored for the specific application, but a general theory is missing. We will develop such a theory in this project.

Facets of Similarity

Similarity of graphs has many different facets. We will identify the common core of different approaches to similarity, but also exhibit their differences.

Methodology

We will design methods for:

• Comparing different similarity measures
• Obtaining a semantic understanding of similarity
• Developing criteria for the suitability of various similarity measures for different types of applications

Focus on Algorithms

A particular focus of our research will be on efficient algorithms for computing similarity. There is little use in having a perfect similarity measure if we have no efficient way of determining how similar two graphs are.

Graph Isomorphism Problem

A classic algorithmic problem in this context is the graph isomorphism problem of deciding whether two graphs are structurally identical. Determining the precise computational complexity of this problem, or of the equivalent problem of computing all symmetries of a graph, is regarded to be one of the most important open questions in theoretical computer science.

Future Directions

Building on recent progress, we will design new algorithms breaking barriers towards a polynomial-time algorithm for the isomorphism problem.