On the multihomogeneous Bézout bound on the number of embeddings of minimally rigid graphs

Abstract

Rigid graph theory is an active area with many open problems, especially regarding embeddings in R d or other manifolds, and tight upper bounds on their number for a given number of vertices. Our premise is to relate the number of embeddings to that of solutions of a well-constrained algebraic system and exploit progress in the latter domain. In particular, the system's complex solutions naturally extend the notion of real embeddings, thus allowing us to employ bounds on complex roots. We focus on multihomogeneous Bézout (m-Bézout) bounds of algebraic systems since they are fast to compute and rather tight for systems exhibiting structure as in our case. We introduce two methods to relate such bounds to combinatorial properties of minimally rigid graphs in C d and S d. The first relates the number of graph orientations to the m-Bézout bound, while the second leverages a matrix permanent formulation. Using these approaches we improve the best known asymptotic upper bounds for planar graphs in dimension 3, and all minimally rigid graphs in dimension d ≥ 5, both in the Euclidean and spherical case. Our computations indicate that m-Bézout bounds are tight for embeddings of planar graphs in S 2 and C 3. We exploit Bernstein's second theorem on the exactness of mixed volume, and relate it to the m-Bézout bound by analyzing the associated Newton polytopes. We reduce the number of checks required to verify exactness by an exponential factor, and conjecture further that it suffices to check a linear instead of an exponential number of cases overall.

The m-Bézout Bound and Distance Geometry

“…This is the approach on which the present work relies. In [3], outdegreeconstrained orientations as well as matrix permanents are related to the m-Bézout bound of certain algebraic systems that compute the embedding number. This work resulted to improved asymptotic upper bounds for d ≥ 5, using the Brégman-Minc permanent bound [9,21].…”

“…More importantly, this work led to the following combinatorial technique. In [4], the target is on a method that bounds the number of outdegree-constrained orientations. It managed to improve the bound on embeddings for all d ≥ 2 (the case of d = 1 is trivial) and proved that the permanent bounds can be ameliorated in that case.…”

“…We exploit the connection between the system's m-Bézout number and the number of constrained orientations of a simple graph that we specify for the systems under investigation, then bound the number of the graph's orientations. This procedure relies on the proofs in [3,4]. It offers the first closed-form bound on m-Bézout numbers; we hope this may prove useful in a fast estimation of the algebraic complexity of problems modeled by multi-homogeneous algebraic equations.…”

“…where Notice that β d < 1, so a n d gives the asymptotic order of this bound. An asymptotic expression of a d is given in [4]:…”

“…In [3], the authors leverage the m-Bézout bound and express it by the number of certain constrained orientations of simple graphs. A combinatorial process on these graphs has recently improved the bound on orientations and, therefore, has improved the bounds on the number of distance graph embeddings [4]. For Laman graphs the new bound is inferior to thus improving upon Bézout's bound for the first time.…”