Computing Heegaard genus is NP-hard

TL;DR

This paper proves that determining whether a 3-manifold has a Heegaard splitting of a given genus is an NP-hard problem, by reducing CNF-SAT to this problem in quadratic time.

Contribution

It establishes the NP-hardness of computing the Heegaard genus of 3-manifolds, a significant complexity result in geometric topology.

Findings

Heegaard genus decision problem is NP-hard.

Reduction from CNF-SAT demonstrates computational difficulty.

Provides a quadratic time reduction method.

Abstract

We show that {\sc Heegaard Genus $\leq g$}, the problem of deciding whether a triangulated 3-manifold admits a Heegaard splitting of genus less than or equal to $g$, is NP-hard. The result follows from a quadratic time reduction of the NP-complete problem {\sc CNF-SAT} to {\sc Heegaard Genus $\leq g$}.