Distance-preserving orderings in graphs

For every connected graph G, a subgraph H of G is isometric if the distance between any two vertices in H is the same in H as in G. A distance-preserving elimination ordering of G is a total ordering of its vertex-set V (G), denoted (v1; v2;...,vn), such that any subgraph Gi = G n (v1; v2;..., vi) with 1 ≤ i < n is isometric. This kind of ordering has been introduced by Chepoi in his study on weakly modular graphs [11]. We prove that it is NP-complete to decide whether such ordering exists for a given graph | even if it has diameter at most 2. Then, we prove on the positive side that the problem of computing a distance-preserving ordering when there exists one is fixed-parameter-tractable in the treewidth. Lastly, we describe a heuristic in order to compute a distance-preserving ordering when there exists one that we compare to an exact exponential time algorithm and to an ILP formulation for the problem.Nous étudions les ordres d’élimination des sommets préservant les distances dans les graphes

Distance-preserving orderings in graphs

For every connected graph G, a subgraph H of G is isometric if the distance between any two vertices in H is the same in H as in G. A distance-preserving elimination ordering of G is a total ordering of its vertex-set V (G), denoted (v1; v2;...,vn), such that any subgraph Gi = G n (v1; v2;..., vi) with 1 ≤ i < n is isometric. This kind of ordering has been introduced by Chepoi in his study on weakly modular graphs [11]. We prove that it is NP-complete to decide whether such ordering exists for a given graph | even if it has diameter at most 2. Then, we prove on the positive side that the problem of computing a distance-preserving ordering when there exists one is fixed-parameter-tractable in the treewidth. Lastly, we describe a heuristic in order to compute a distance-preserving ordering when there exists one that we compare to an exact exponential time algorithm and to an ILP formulation for the problem.Nous étudions les ordres d’élimination des sommets préservant les distances dans les graphes

On distance-preserving elimination orderings in graphs: Complexity and algorithms

International audienceFor every connected graph G, a subgraph H of G is isometric if the distance between any two vertices in H is the same in H as in G. A distance-preserving elimination ordering of G is a total ordering of its vertex-set V (G), denoted (v 1 , v 2 ,. .. , v n), such that any subgraph G i = G\(v 1 , v 2 ,. .. , v i) with 1 ≤ i < n is isometric. This kind of ordering has been introduced by Chepoi in his study on weakly modular graphs [11]. We prove that it is NP-complete to decide whether such ordering exists for a given graph — even if it has diameter at most 2. Then, we prove on the positive side that the problem of computing a distance-preserving ordering when there exists one is fixed-parameter-tractable in the treewidth. Lastly, we describe a heuristic in order to compute a distance-preserving ordering when there exists one that we compare to an exact exponential time algorithm and to an ILP formulation for the problem

Discretization vertex orders in distance geometry

CNPQ - CONSELHO NACIONAL DE DESENVOLVIMENTO CIENTÍFICO E TECNOLÓGICOCAPES - COORDENAÇÃO DE APERFEIÇOAMENTO DE PESSOAL DE NÍVEL SUPERIORFAPESP - FUNDAÇÃO DE AMPARO À PESQUISA DO ESTADO DE SÃO PAULOWhen a weighted graph is an instance of the Distance Geometry Problem (DGP), certain types of vertex orders (called discretization orders) allow the use of a very efficient, precise and robust discrete search algorithm (called Branch-and-Prune). Accordingly, finding such orders is critically important in order to solve DGPs in practice. We discuss three types of discretization orders, the complexity of determining their existence in a given graph, and the inclusion relations between the three order existence problems. We also give three mathematical programming formulations of some of these ordering problems.When a weighted graph is an instance of the Distance Geometry Problem (DGP), certain types of vertex orders (called discretization orders) allow the use of a very efficient, precise and robust discrete search algorithm (called Branch-and-Prune). Accordingly, finding such orders is critically important in order to solve DGPs in practice. We discuss three types of discretization orders, the complexity of determining their existence in a given graph, and the inclusion relations between the three order existence problems. We also give three mathematical programming formulations of some of these ordering problems.1972741CNPQ - CONSELHO NACIONAL DE DESENVOLVIMENTO CIENTÍFICO E TECNOLÓGICOCAPES - COORDENAÇÃO DE APERFEIÇOAMENTO DE PESSOAL DE NÍVEL SUPERIORFAPESP - FUNDAÇÃO DE AMPARO À PESQUISA DO ESTADO DE SÃO PAULOCNPQ - CONSELHO NACIONAL DE DESENVOLVIMENTO CIENTÍFICO E TECNOLÓGICOCAPES - COORDENAÇÃO DE APERFEIÇOAMENTO DE PESSOAL DE NÍVEL SUPERIORFAPESP - FUNDAÇÃO DE AMPARO À PESQUISA DO ESTADO DE SÃO PAULOsem informaçãosem informaçãosem informaçã

Finding Optimal Discretization Orders For Molecular Distance Geometry By Answer Set Programming

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)The Molecular Distance Geometry Problem (MDGP) is the problem of finding the possible conformations of a molecule by exploiting available information about distances between some atom pairs. When particular assumptions are satisfied, the MDGP can be discretized, so that the search domain of the problem becomes a tree. This tree can be explored by using an interval Branch & Prune (iBP) algorithm. In this context, the order given to the atoms of the molecules plays an important role. In fact, the discretization assumptions are strongly dependent on the atomic ordering, which can also impact the computational cost of the iBP algorithm. In this work, we propose a new partial discretization order for protein backbones. This new atomic order optimizes a set of objectives, that aim at improving the iBP performances. The optimization of the objectives is performed by Answer Set Programming (ASP), a declarative programming language that allows to express our problem by a set of logical constraints. The comparison with previously proposed orders for protein backbones shows that this new discretization order makes iBP perform more efficiently. © Springer International Publishing Switzerland 2016.610115CNPq, Conselho Nacional de Desenvolvimento Científico e TecnológicoConselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)Berman, H.M., Westbrook, J., Feng, Z., Gilliland, G., Bhat, T.N., Weissig, H., Shindyalov, I.N., Bourne, P.E., The protein data bank (2000) Nucleic Acid Res, 28, pp. 235-242Brewka, G., Eiter, T., Truszczyński, M., Answer set programming at a glance (2011) Commun. ACM, 54 (12), pp. 92-103Cassioli, A., Bardiaux, B., Bouvier, G., Mucherino, A., Alves, R., Liberti, L., Nilges, M., Malliavin, T.E., An algorithm to enumerate all possible protein conformations verifying a set of distance restraints (2015) BMC Bioinform, , (, to appear)Cassioli, A., Gunluk, O., Lavor, C., Liberti, L., Discretization vertex orders in distance geometry (2015) Discrete Appl. Math, , (, to appear)Costa, V., Mucherino, A., Lavor, C., Cassioli, A., Carvalho, L.M., Maculan, N., Discretization orders for protein side chains (2014) J. Glob. Optim, 60 (2), pp. 333-349Crippen, G.M., Havel, T.F., (1988) DistanceGeometry and Molecular Conformation, , (Wiley, NewYork,)Eiter, T., Ianni, G., Krennwallner, T., Answer set programming: A primer (2009) Reason. Web, 5689, pp. 40-110Gebser, M., Kaufmann, B., Schaub, T., Conflict-driven answer set solving: From theory to practice (2012) Artif. Intell, 187, pp. 52-89Gelfond, M., Answer Sets (2007) Handbook of Knowledge Representation, , Chapter 7 (Elsevier, Amsterdam,)Gonçalves, D.S., Mucherino, A., Discretization orders and efficient computation of Cartesian coordinates for distance geometry (2014) Optim. Lett, 8 (7), pp. 2111-2125Gramacho, W., Gonçalves, D., Mucherino, A., Maculan, N., A new algorithm to finding discretizable orderings for distance geometry (2013) Proceedings of Distance Geometry and Applications (DGA13), pp. 149-152. , Manaus, Amazonas, BrazilHavel, T.F., Distance Geometry (1995) Encyclopedia of nuclear magnetic resonance, pp. 1701-1710. , ed. by D.M. Grant, R.K. Harris (Wiley, New York,)Lavor, C., Lee, J., Lee-St John, A., Liberti, L., Mucherino, A., Sviridenko, M., Discretization orders for distance geometry problems (2012) Optim. Lett, 6 (4), pp. 783-796Lavor, C., Liberti, L., Maculan, N., Mucherino, A., The discretizable molecular distance geometry problem (2012) Comput. Optim. Appl, 52, pp. 115-146Lavor, C., Liberti, L., Mucherino, A., The interval branch-and-prune algorithm for the discretizable molecular distance geometry problem with inexact distances (2013) J. Glob. Optim, 56 (3), pp. 855-871Lavor, C., Mucherino, A., Liberti, L., Maculan, N., On the computation of protein backbones by using artificial backbones of hydrogens (2011) J. Glob. Optim, 50 (2), pp. 329-344Liberti, L., Lavor, C., Maculan, N., A branch-and-prune algorithm for the molecular distance geometry problem (2008) Int. Trans. Oper. Res, 15, pp. 1-17Liberti, L., Lavor, C., Maculan, N., Mucherino, A., Euclidean distance geometry and applications (2014) SIAM Rev, 56 (1), pp. 3-69Malliavin, T.E., Mucherino, A., Nilges, M., Distance geometry in structural biology: New perspectives (2013) Distance Geometry: Theory, pp. 329-350. , Methods and Applications, ed. by A. Mucherino, C. Lavor, L. Liberti, N. Maculan (Springer, Berlin,)Mucherino, A., On the Identification of Discretization Orders for Distance Geometry with Intervals, Lecture Notes in Computer Science 8085 (2013) Proceedings of Geometric Science of Information (GSI13), pp. 231-238. , ed. by F. Nielsen, F. Barbaresco, Paris, FranceMucherino, A., APseudo de Bruijn GraphRepresentation for DiscretizationOrders for Distance Geometry, LectureNotes in Computer Science 9043, LectureNotes in Bioinformatics series (2015) Proceedings of the 3rd InternationalWork-Conference on Bioinformatics and Biomedical Engineering (IWBBIO15), pp. 514-523. , ed. by F. Ortuño, I. Rojas Granada, SpainMucherino, A., Lavor, C., Liberti, L., The discretizable distance geometry problem (2012) Optim. Lett, 6 (8), pp. 1671-1686Ramachandran, G.N., Ramakrishnan, C., Sasisekharan, V., Stereochemistry of polypeptide chain conformations (1963) J. Mol. Biol, 7, pp. 95-99Saxe, J.B., Embeddability of weighted graphs in k-space is strongly NP-hard (1979) Proceedings of 17th Allerton Conference in Communications, Control and Computing, pp. 480-48