On the Counting Complexity of Mathematical Nanosciences

We investigate the algorithmic hardness of a series of counting problems that come from crystal physics and fullerene chemistry. We claim that those problems are representative of mathematical nanosciences, and we observe that all them are sparse. It follows from Mahaney's work that sparse problems cannot be hard for NP. Then, we have to use a different complexity class to analyze the aforementioned (seemingly hard) problems. We study the complexity class #P-1, which is constituted by all the tally counting problems that belong to the counting class #P. We conjecture that counting matchings in square grids, counting Hamiltonian cycles in square grids, and counting Clar sets in fullerene graphs are all hard for #P-1. We prove some weak results related to these conjectures. We also consider the restriction of these three problems to carbon nanotubes, and we prove that those restrictions can be solved in logarithmic time. We get these tractability results about nanotubes as corollaries of a Buchi-like Theorem for linear lattices.

Automated summary

This study investigates the algorithmic hardness of counting problems in crystal physics and fullerene chemistry, claiming them to be representative of mathematical nanosciences and observing their sparsity. By analyzing the complexity class #P-1, it is concluded that these seemingly hard problems cannot be hard for NP. Additionally, conjectures and weak results related to counting matchings, Hamiltonian cycles, and Clar sets are discussed in the paper.