463 research outputs found
Mixed Volume and Distance Geometry Techniques for Counting Euclidean Embeddings of Rigid Graphs
A graph G is called generically minimally rigid in Rd if, for any choice of sufficiently generic edge lengths, it can be embedded in Rd 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 R2 and R3, 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 ≤ 10 in R2 and R3, which reduce the existing gaps and lead to tight bounds for n ≤ 7 in both R2 and R3; in particular, we describe the recent settlement of the case of Laman graphs with 7 vertices. Our approach also yields a new upper bound for Laman graphs with 8 vertices, which is conjectured to be tight. We also establish the first lower bound in R3 of about 2.52n, where n denotes the number of vertices
Mixed volume and distance geometry techniques for counting Euclidean embeddings of rigid graphs
International audienceA graph is called generically minimally rigid in \RR^d if, for any choice of sufficiently generic edge lengths, it can be embedded in \RR^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, 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 \RR^2 and \RR^3, where Laman graphs and 1-skeleta of convex simplicial polyhedra, respectively, admit inductive Henneberg constructions. Our implementation yields upper bounds for in \RR^2 and \RR^3, which reduce the existing gaps and lead to tight bounds for in both \RR^2 and \RR^3; in particular, we describe the recent settlement of the case of Laman graphs with 7 vertices. We also establish the first lower bound in \RR^3 of about , where denotes the number of vertices
Lower bounds on the number of realizations of rigid graphs
Computing the number of realizations of a minimally rigid graph is a
notoriously difficult problem. Towards this goal, for graphs that are minimally
rigid in the plane, we take advantage of a recently published algorithm, which
is the fastest available method, although its complexity is still exponential.
Combining computational results with the theory of constructing new rigid
graphs by gluing, we give a new lower bound on the maximal possible number of
(complex) realizations for graphs with a given number of vertices. We extend
these ideas to rigid graphs in three dimensions and we derive similar lower
bounds, by exploiting data from extensive Gr\"obner basis computations
On the maximal number of real embeddings of minimally rigid graphs in , and
Rigidity theory studies the properties of graphs that can have rigid
embeddings in a euclidean space or on a sphere and which in
addition satisfy certain edge length constraints. One of the major open
problems in this field is to determine lower and upper bounds on the number of
realizations with respect to a given number of vertices. This problem is
closely related to the classification of rigid graphs according to their
maximal number of real embeddings.
In this paper, we are interested in finding edge lengths that can maximize
the number of real embeddings of minimally rigid graphs in the plane, space,
and on the sphere. We use algebraic formulations to provide upper bounds. To
find values of the parameters that lead to graphs with a large number of real
realizations, possibly attaining the (algebraic) upper bounds, we use some
standard heuristics and we also develop a new method inspired by coupler
curves. We apply this new method to obtain embeddings in . One of
its main novelties is that it allows us to sample efficiently from a larger
number of parameters by selecting only a subset of them at each iteration.
Our results include a full classification of the 7-vertex graphs according to
their maximal numbers of real embeddings in the cases of the embeddings in
and , while in the case of we achieve this
classification for all 6-vertex graphs. Additionally, by increasing the number
of embeddings of selected graphs, we improve the previously known asymptotic
lower bound on the maximum number of realizations. The methods and the results
concerning the spatial embeddings are part of the proceedings of ISSAC 2018
(Bartzos et al, 2018)
Mixed Volume Techniques for Embeddings of Laman Graphs
Determining the number of embeddings of Laman graph frameworks is an open
problem which corresponds to understanding the solutions of the resulting
systems of equations. In this paper we investigate the bounds which can be
obtained from the viewpoint of Bernstein's Theorem. The focus of the paper is
to provide the methods to study the mixed volume of suitable systems of
polynomial equations obtained from the edge length constraints. While in most
cases the resulting bounds are weaker than the best known bounds on the number
of embeddings, for some classes of graphs the bounds are tight.Comment: Thorough revision of the first version. (13 pages, 4 figures
New upper bounds for the number of embeddings of minimally rigid graphs
By definition, a rigid graph in (or on a sphere) has a finite
number of embeddings up to rigid motions for a given set of edge length
constraints. These embeddings are related to the real solutions of an algebraic
system. Naturally, the complex solutions of such systems extend the notion of
rigidity to . A major open problem has been to obtain tight upper
bounds on the number of embeddings in , for a given number
of vertices, which obviously also bound their number in .
Moreover, in most known cases, the maximal numbers of embeddings in
and coincide. For decades, only the trivial bound
of was known on the number of embeddings.Recently, matrix
permanent bounds have led to a small improvement for . This work
improves upon the existing upper bounds for the number of embeddings in
and , by exploiting outdegree-constrained orientations on a
graphical construction, where the proof iteratively eliminates vertices or
vertex paths. For the most important cases of and , the new bounds
are and , respectively. In general, the
recent asymptotic bound mentioned above is improved by a factor of . Besides being the first substantial improvement upon a long-standing
upper bound, our method is essentially the first general approach relying on
combinatorial arguments rather than algebraic root counts
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