Counting (with) homomorphisms

Introduction

This project addresses the computational complexity of counting problems in graphs; such problems ask to compute the number of certain structures rather than merely deciding their existence.

Applications of Counting Problems

Counting problems find applications in diverse areas like:

• Network analysis
• Machine learning
• Probabilistic databases
• Statistical physics

They are linked to fundamental questions in complexity theory and give rise to algorithmic breakthroughs for decision problems.

Project Goals

The proposed project will go beyond the state of the art in computational counting by building bridges between algorithms/complexity and the mathematical theory of graph homomorphisms, which are structure-preserving maps between graphs.

Historical Context

Already starting from the 1960s, Lovász and others showed that homomorphism numbers between graphs tie together disparate mathematical areas. Similarly, the PI observed in the past 5 years that the computational problem of counting homomorphisms unifies a range of problems in computer science that were previously studied in isolation.

Research Implications

This connection allows us to approach fundamental questions in:

1. Parameterized complexity
2. Fine-grained complexity

These questions relate to the complexity of counting small patterns in large graphs, which were out of reach for more combinatorial approaches.

Revisiting Classical Problems

Secondly, it allows us to revisit problems in classical counting complexity from a new angle, especially for partition functions in discrete physical systems, with the aim of simplifying, unifying, and extending known results.

Algebraic Complexity Perspective

Thirdly, homomorphism counts give a surprising novel viewpoint on questions in algebraic complexity surrounding the “permanent versus determinant” problem, an algebraic variant of the “P versus NP” problem.