A d-dimensional framework is a straight line realization of a graph G in R d. We shall only consider generic frameworks, in which the co-ordinates of all the vertices of G are algebraically independent. Two frameworks for G are equivalent if corresponding edges in the two frameworks have the same length. A framework is a unique realization of G in R d if every equivalent framework can be obtained from it by an isometry of R d. Bruce Hendrickson proved that if G has a unique realization in R d then G is (d + 1)-connected and redundantly rigid. He conjectured that every realization of a (d + 1)connected and redundantly rigid graph in R d is unique. This conjecture is true for d = 1 but was disproved by Robert Connelly for d ≥ 3. We resolve the remaining open case by showing that Hendrickson’s conjecture is true for d = 2. As a corollary we deduce that every realization of a 6-connected graph as a 2-dimensional generic framework is a unique realization. Our proof is based on a new inductive characterization of 3-connected graphs whose rigidity matroid is connected.

. The molecule problem is that of determining the relative locations of a set of objects in Euclidean space relying only upon a sparse set of pairwise distance measurements. This NP--hard problem has applications in the determination of molecular conformation. The molecule problem can be naturally expressed as a continuous, global optimization problem, but it also has a rich combinatorial structure. This paper investigates how that structure can be exploited to simplify the optimization problem. In particular, we present a novel divide--and--conquer algorithm in which a large global optimization problem is replaced by a sequence of smaller ones. Since the cost of the optimization can grow exponentially with problem size, this approach holds the promise of a substantial improvement in performance. Our algorithmic development relies upon some recently published results in graph theory. We describe an implementation of this algorithm and report some results of its performance on a sample ...

A subspace adaptation of the Coleman-Li trust region and interior method is proposed for solving large-scale bound-constrained minimization problems. This method can be implemented with either sparse Cholesky factorization or conjugate gradient computation. Under reasonable conditions the convergence properties of this subspace trust region method are as strong as those of its full-space version.

We show that a continuation approach to global optimization with global smoothing techniques can be used to obtain "-optimal solutions to distance geometry problems. We show that determining an "-optimal solution is still an NP-hard problem when " is small. A discrete form of the Gaussian transform is proposed based on the Hermite form of Gaussian quadrature. We show that the modified transform can be used whenever the transformed functions cannot be computed analytically. Our numerical results show that the discrete Gauss transform can be used to obtain "-optimal solutions for general distance geometry problems, and in particular, to determine the three-dimensional structure of protein fragments.

A number of current technologies allow for the determination of inter-atomic distance information in structures such as proteins and RNA. Thus, the reconstruction of a three-dimensional set of points using information about its inter-point distances has become a task of basic importance in determining molecular structure. The distance measurements one obtains from techniques such as NMR are typically sparse and error-prone, greatly complicating the reconstruction task. Many of these errors result in distance measurements that can be safely assumed to lie within certain fixed tolerances. But a number of sources of systematic error in these experiments lead to inaccuracies in the data that are very hard to quantify; in effect, one must treat certain entries of the measured distance matrix as being arbitrarily "corrupted." The existence of arbitrary errors leads to an interesting sort of error--correction problem --- how many corrupted entries in a distance matrix can be efficiently corre...

Abstract. We study the performance of the dgsol code for the solution of distance geometry problems with lower and upper bounds on distance constraints. The dgsol code uses only a sparse set of distance constraints, while other algorithms tend to work with a dense set of constraints either by imposing additional bounds or by deducing bounds from the given bounds. Our computational results show that protein structures can be determined by solving a distance geometry problem with dgsol and that the approach based on dgsol is significantly more reliable and efficient than multi-starts with an optimization code.

A multi-graph G on n vertices is (k,ℓ)-sparse if every subset of n ′ ≤ n vertices spans at most kn ′ − ℓ edges. G is tight if, in addition, it has exactly kn − ℓ edges. For integer values k and ℓ ∈ [0,2k), we characterize the (k,ℓ)-sparse graphs via a family of simple, elegant and efficient algorithms called the (k,ℓ)-pebble games.

AMS Subject Classification: The sensor network localization, SNL, problem in embedding dimension r, consists of locating the positions of wireless sensors, given only the distances between sensors that are within radio range and the positions of a subset of the sensors (called anchors). Current solution techniques relax this problem to a weighted, nearest, (positive) semidefinite programming, SDP,completion problem, by using the linear mapping between Euclidean distance matrices, EDM, and semidefinite matrices. The resulting SDP is solved using primal-dual interior point solvers, yielding an expensive and inexact solution. This relaxation is highly degenerate in the sense that the feasible set is restricted to a low dimensional face of the SDP cone, implying that the Slater constraint qualification fails. Cliques in the graph of the SNL problem give rise to this degeneracy in the SDP relaxation. In this paper, we take advantage of the absence of the Slater constraint qualification and derive a technique for the SNL problem, with exact data, that explicitly solves the corresponding rank restricted SDP problem. No SDP solvers are used. For randomly generated instances,

... In this article we survey some results and provide references about these problems for the following matrix properties: positive semidefinite matrices, Euclidean distance matrices, completely positive matrices, contraction matrices, and matrices of given rank. We treat mainly optimization and combinatorial aspects.

Survey article for the proceedings of Discrete Optimization '99 where some of these results were presented as a plenary address. y