Computational Hardness of Enumerating Satisfying Spin-Assignments in Triangulations

TL;DR

This paper proves that determining the existence and counting the number of satisfying spin-assignments in surface triangulations is computationally hard, specifically NP-complete and #P-complete, via a complex reduction from Boolean formulas.

Contribution

It establishes the computational complexity of finding and counting satisfying spin-assignments in surface triangulations, linking it to well-known hard problems.

Findings

Deciding existence of satisfying spin-assignments is NP-complete.

Counting the number of satisfying spin-assignments is #P-complete.

Reductions are based on mapping Boolean formulas to surface triangulations.

Abstract

Satisfying spin-assignments in triangulations of a surface are states of minimum energy of the antiferromagnetic Ising model on triangulations which correspond (via geometric duality) to perfect matchings in cubic bridgeless graphs. In this work we show that it is NP-complete to decide whether or not a surface triangulation admits a satisfying spin-assignment, and that it is #P-complete to determine the number of such assignments. Both results are derived via an elaborate (and atypical) reduction that maps a Boolean formula in 3-conjunctive normal form into a triangulation of an orientable closed surface.