Algorithms and complexity for Turaev-Viro invariants

TL;DR

This paper analyzes the computational complexity of Turaev-Viro invariants for 3-manifolds, proving polynomial-time computability for r=3, -hardness for r=4, and providing a practical fixed-parameter tractable algorithm for all r.

Contribution

It establishes the complexity classifications for r=3 and r=4, and introduces an efficient fixed-parameter tractable algorithm for computing Turaev-Viro invariants.

Findings

Computing Turaev-Viro invariants for r=3 is polynomial time.

Computing for r=4 is -hard.

The proposed algorithm is practical and outperforms previous methods.

Abstract

The Turaev-Viro invariants are a powerful family of topological invariants for distinguishing between different 3-manifolds. They are invaluable for mathematical software, but current algorithms to compute them require exponential time. The invariants are parameterised by an integer $r \geq 3$. We resolve the question of complexity for $r=3$ and $r=4$, giving simple proofs that computing Turaev-Viro invariants for $r=3$ is polynomial time, but for $r=4$ is \#P-hard. Moreover, we give an explicit fixed-parameter tractable algorithm for arbitrary $r$, and show through concrete implementation and experimentation that this algorithm is practical---and indeed preferable---to the prior state of the art for real computation.