Mixed Volume and Distance Geometry Techniques for Counting Euclidean Embeddings of Rigid Graphs

Abstract

A graph G is called generically minimally rigid in ℝ ^d if, for any choice of sufficiently generic edge lengths, it can be embedded in ℝ ^d in a finite number of distinct ways, modulo rigid transformations. Here, we deal with the problem of determining tight bounds on the number of such embeddings, as a function of the number of vertices. The study of rigid graphs is motivated by numerous applications, mostly in robotics, bioinformatics, sensor networks, and architecture. We capture embeddability by polynomial systems with suitable structure so that their mixed volume, which bounds the number of common roots, yields interesting upper bounds on the number of embeddings. We explore different polynomial formulations so as to reduce the corresponding mixed volume, namely by introducing new variables that remove certain spurious roots and by applying the theory of distance geometry. We focus on ${\mathbb{R}}^{2}$ and ${\mathbb{R}}^{3}$, where Laman graphs and 1-skeleta (or edge graphs) of convex simplicial polyhedra, respectively, admit inductive Henneberg constructions. Our implementation yields upper bounds for $n \leq 10$ in ${\mathbb{R}}^{2}$ and ${\mathbb{R}}^{3}$, which reduce the existing gaps and lead to tight bounds for $n \leq 7$ in both ${\mathbb{R}}^{2}$ and ${\mathbb{R}}^{3}$; in particular, we describe the recent settlement of the case of Laman graphs with seven vertices. Our approach also yields a new upper bound for Laman graphs with eight vertices, which is conjectured to be tight. We also establish the first lower bound in ${\mathbb{R}}^{3}$ of about 2. 52ⁿ, where n denotes the number of vertices.