J. B. Saxe. Embeddability of weighted graphs in k-space is strongly NP-hard. Proc. 17th Allerton Conf. in Communications, Control, and Computing, pages 480-489, 1979.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....of Science Exchange Programme. Department of Operations Research, E otv os University, P azm any P eter s et any 1 C, 1117 Budapest, Hungary. e mail:jordan cs.elte.hu Supported by the Hungarian Scienti c Research Fund grant no. T037547, F034930, and FKFP grant no. 0143 2001. is unique. Saxe [14] proved that this problem is NP hard. We obtain a problem of di erent type, however, if we exclude degenerate cases. A framework (G; p) is said to be generic if the coordinates of all the points are algebraically independent over the rationals. In what follows we shall consider the unique ....

J.B. Saxe, Embeddability of weighted graphs in k-space is strongly NP-hard, Tech. Report, Computer Science Department, Carnegie-Mellon University, Pittsburgh, PA, 1979.

....output is a three dimensional structure which satisfies those distance constraints as well as possible. In order to incorporate a piece of structural data into a DG calculation, that information needs to be translated into some distance bounds. This specialized problem, although NP hard in general [83], is rich in combinatorial structure and opportunities for optimization, and there are several elegant mathematical theories for solving the DG problem. Given a DG problem, does it have a unique solution (up to translations, rotations, and mirror images) assuming highly precise input distances ....

....and mirror images) assuming highly precise input distances The answer is of interest because it can guide us regarding the amount of data to collect and aid in the structure computation itself. Unfortunately, this decision is just as difficult as the DG problem itself, NP hard in general [83]. However, because an n point structure in three space has (3n 6) degrees of freedom, a necessary condition is at least that many distance constraints. Moreover, 2] offers a simple sufficient condition for unique solution. First, any four points can be determined by specifying all six ....

J. B. Saxe, "Embeddability of Weighted Graphs in k-Space Is Strongly NP-Hard", pp. 480489, in Proc. 17th Allerton Conference in Communications, Control, and Computing, Monticello, IL, 1979.

....fill in [La98d] as in the psd case, this follows from the fact (mentioned below) that partial matrices that are completable to a distance matrix admit a good characterization when their pattern is a chordal graph. While the exact complexity of problem (EDM) is not known, it has been shown in [Sa79] that problem (EDMk) is NP complete if k = 1 and NP hard if k 2 (even when restricted to partial matrices with entries in f1; 2g) Finding ffl optimal solutions to the graph realization problem is also NP hard for small ffl ( MW96] The graph realization problem (EDMk) has been much studied, in ....

....In [He90, He95] the basic step consist of finding principal submatrices having a unique realization, treating each of them separately and then trying to combine the solutions. Thus arises the problem of identifying principal submatrices having a unique realization, which turns out to be NP hard [Sa79]. However, several necessary conditions for unicity of realization are known, related with connectivity and generic rigidity properties of the graph pattern [Ye79, He92] Generic rigidity of graphs can be characterized and recognized in polynomial time only in dimension k 2 ( La70] LY82] cf. ....

J.B. Saxe. Embeddability of weighted graphs in k-space is strongly NP-hard. Proceedings of the 17th Allerton Conference in Communications, Control and Computing, pages 480--489, 1979.

....the three dimensional Euclidean space E 3 so that any two nodes with an edge between them are mapped to points whose Euclidean distance equals the weight of that edge. The above problem is in fact NP hard in the strong sense, and it was one of the rst geometric problems shown to be in this class [19]. There are only few other such examples of geometric problems, e.g. the Steiner Minimal Tree problem. The focus of this paper is on constructive techniques for the question of embedding, with little reference to complexity issues. The problem of identifying the conformation of globular proteins ....

J. B. Saxe. Embeddability of weighted graphs in k-space is strongly NP-hard. In Proc. 17th Allerton Conf. on Communications, Control and Computing, pages 480489, 1979.

....problem, when all exact distances are given, can be solved in polynomial time and is a tractable problem. 2.3 Problems with Sparse Sets of Distances In practice, we are often given only a sparse set of distances. The distance geometry problem then becomes very hard to solve in general. Saxe [16] in 1979 showed that a one dimensional distance geometry problem is equivalent to a set partition problem which is known to be NP hard. He also showed that a N dimensional distance geometry problem is NP hard for all N greater or equal to one. NP hard is a computational complexity term. A ....

J. B. Saxe, Embeddability of Weighted Graphs in K-Space Is Strongly NP-Hard, in Proc. 17th Allerton Conference in Communications, Control and Computing, 1979, pp. 480--489.

....solving a distance geometry problem using the NMR distance data. They can be formulated as global nonlinear least squares optimization problems (Crippen and Havel [30] or, from a graph theoretic point of view, 10 they are a class of NP complete graph embedding problems (Hendrickson [41] Saxe [79, 80], Mor e and Wu [64] Recent attempts to solve these problems on parallel high performance architectures are by (Hendrickson [42] Mor e and Wu [66] Byrd, Schnabel et al. [97] etc) A simple version of the distance geometry problem is to find a set of points to realize a given set of distances ....

J. B. Saxe. Embeddability of weighted graphs in k-space is strongly NPhard. In Proc. 17th Allerton Conference in Communications, Controli and Computing, pages 480--489, 1979.

....solving a distance geometry problem using the NMR distance data. They can be formulated as global nonlinear least squares optimization problems (Crippen and Havel [30] or, from a graph theoretic point of view, 10 they are a class of NP complete graph embedding problems (Hendrickson [41] Saxe [79, 80], Mor e and Wu [64] Recent attempts to solve these problems on parallel high performance architectures are by (Hendrickson [42] Mor e and Wu [66] Byrd, Schnabel et al. [97] etc) A simple version of the distance geometry problem is to find a set of points to realize a given set of distances ....

J. B. Saxe. Embeddability of weighted graphs in k-space is strongly NP-hard. Technical report, Department of Computer Science, CarnegieMellon University, Pittsburgh, PA, 1979.

....A simple objective function can be defined to enforce the constraints. This optimization problem is believed to be computationally intractable in general because it has been shown to be strongly NP complete in the one dimensional case [6, 13] and strongly NP hard in the higher dimension case [14]. A large number of such methods for solving distance geometry problems have been proposed, such as Crippen and Havel[6] Havel[8] Hendrickson[9] Glunt, Hayden, Raydan[7] and Mor e and Wu[12, 13] The method we present in this paper is based on the stochastic perturbation global optimization ....

....a solution x is equivalent to a global minimizer of the function (2.5) with function value 0. The difficulty in solving the distance geometry problem arises from several sources. First, the problem in itself is strongly NP complete in one 4 dimension, and strongly NP hard in higher dimension[6, 14, 13], therefore it is very unlikely that an efficient algorithm for solving all cases of the problem can be found. Also there are huge numbers of local minimizers for the functions (2.4) and (2.5) which makes it very challenging to locate the basin of attraction of the global minimizer. It can be ....

J.B.Saxe, Embeddability of Weighted Graphs in k-space is Strongly NPhard, Proc. 17th Allerton Conference in Communications, Control and Computing, 1979, pp. 480-489.

....genetics, and geography [1] In some applications (e.g. the molecular conformation problems in chemistry [4, 15] the embedding dimension r is required to be 2 or 3, while in others (e.g. multidimensional scaling in statistics [12, 13] only a small r is desired. It was shown by Saxe [20] that for any given positive integer r, the r embeddability problem of an integerweighted graph G = V; E; is NP Hard. It remains so even if the weights ij are restricted to 1 or 2 only. In [3] Barvinok proved that if G = V; E; is embeddable, then it is r embeddable where r = b( ....

J. B. SAXE. Embeddability of weighted graphs in k-space is strongly NP-hard. Proc. 17th Allerton Conf. in Communications, Control, and Computing, pages 480--489, 1979.

....to determine the three dimensional shape of the molecule, which is important in understanding the molecule s properties. Unfortunately, the graph realization problem is known to be difficult. Saxe has shown it to be strongly NP complete in one dimension and strongly NP hard for higher dimensions [35] . In practice, this means that one is unlikely to find an efficient general algorithm to solve it. However, the graphs and edge lengths that Saxe uses in his proofs are very special and are highly unlikely to occur in practical problems. This paper will address a closely related problem: when ....

J. B. Saxe, Embeddability of weighted graphs in k-space is strongly NP-hard, in Proc. 17th Allerton Conference in Communications, Control and Computing, 1979, pp. 480--489.

....of possible inconsistencies in the experimental data. The distance geometry problem (2. 1) is computationally intractable because the restriction of the distance geometry problem to a one dimensional space is equivalent to the set partition problem, which is known to be NP complete [10] Saxe [42] shows that k dimensional distance geometry problems are strongly NP hard for all k 1. The following result of Mor e and Wu [34] shows that obtaining an approximate solution to the distance geometry problem is also NP hard. Theorem 2.1 Determining an optimal solution to the distance geometry ....

J. B. Saxe, Embeddability of weighted graphs in k-space is strongly NP-hard, in Proc. 17th Allerton Conference in Communications, Control and Computing, 1979, pp. 480-- 489.

....is known the work most closely associated with problems of reconstruction is that dealing with global rigidity [4, 9, 10] Here, the issue is to determine when a set of exact distances is sufficient to uniquely determine a point set up to congruence. Finally, we note related work of Saxe [15], proving that it is NP hard to decide whether a sufficiently sparse set of distances is consistent with a set of points in R d . This contrasts with the results of Section 2, which imply that if the set of distances is sufficiently dense, then the set of all consistent conformations can be ....

J. Saxe, "Embeddability of weighted graphs in k-space is strongly NP-hard," Proc. 19th Allerton Conf. on Computers, Controls, and Communications, 1979, pp. 480--489.

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J. B. Saxe. Embeddability of weighted graphs in k-space is strongly NP-hard. Proc. 17th Allerton Conf. in Communications, Control, and Computing, pages 480-489, 1979.

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J. B. Saxe. Embeddability of weighted graphs in k-space is strongly NP-hard. In Proc. 17th Allerton Conf. Commun. Control Comput., pages 480--489, 1979.

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Saxe, J. B. 1979. Embeddability of weighted graphs in k-space is strongly np-hard. In In Proc. 17th Alleron Conf. Commun. Control Comput., 480--489.

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J. B. Saxe. Embeddability of weighted graphs in k-space is strongly NP-hard. In Proc. 17th Allerton Conf. Commun. Control Comput., pages 480--489, 1979.

No context found.

Saxe, J.B. (1979), Embeddability of weighted graphs in k-space is strongly NP-hard, Proc. of 17th Allerton Conference in Communications, Control, and Computing, Monticello, IL, 480-489.

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Saxe, J. B. Embeddability of weighted graphs in k-space is strongly NP-hard. In Proc. 17th Allerton Conf. Commun. Control Comput. (1979), pp. 480--489.

No context found.

J.B. Saxe, "Embeddability of weighted graphs in k-space is strongly NP-hard," in Proceedings of the 17th Allerton Conference in Communications, Control and Computing, 1979, pp. 480--489.

No context found.

J.B. Saxe, "Embeddability of weighted graphs in k-space is strongly NP-hard," in Proceedings of the 17th Allerton Conference in Communications, Control and Computing, 1979, pp. 480--489.

No context found.

J.B. Saxe. Embeddability of weighted graphs in k-space is strongly NP-hard. In Proceedings of the 17th Allerton Conference in Communications, Control and Computing, pages 480--489, 1979.

No context found.

J.B. SAXE. Embeddability of weighted graphs in k-space is strongly NP-hard. In Seventeenth Annual Allerton Conference on Communication, Control, and Computing, Proceedings of the Conference held in Monticello, Ill., October 10--12, 1979.

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J. B. Saxe, Embeddability of weighted graphs in k-space is strongly NP-hard, Proc. 17th Allerton Conf. in Communications, Control, and Computing, (1979), pp. 480--489.

No context found.

J.B. Saxe. Embeddability of weighted graphs in k-space is strongly NP-hard. In Proceedings of the 17th Allerton Conference in Communications, Control and Computing, pages 480--489, 1979.

No context found.

J.B. Saxe. Embeddability of weighted graphs in k-space is strongly NP-hard. In Proceedings of the 17th Allerton Conference in Communications, Control and Computing, pages 480--489, 1979.

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