Upper and Lower Bounds on the Complexity of Unit-Distance Graph Realization in $\mathbb{Z}^d$

Abstract

It is known that the problem of recognizing the existence of a unit-distance realization in $\mathbb{Z}^d$ for a graph $G$ is NP-complete. Here, assuming ETH, we give a sub-exponential lower bound of $2^{Ω(\sqrt{n})}$ for this problem in $\mathbb{Z}^2$ and $2^{Ω(n^{2/3})}$ in $\mathbb{Z}^3$, and give an upper bound of $2^{O(n^{(d-1)/d} (\log n)^2)}$ for the time complexity with an explicit divide-and-conquer algorithm. This approach beats naive exhaustive search by a factor of $\tildeΩ(n^{1/d})$ in the exponent. For 2 and 3 dimensions, our upper and lower bounds match to within a logarithmic factor.