Experimentos computacionales en la resolución del problema de códigos de identificación

El Problema de Códigos de Identificación (PCI) es un problema NP-difícil relativamente nuevo que, además de contar con aplicaciones concretas (véase el trabajo de Karpovsky, Chakrabarty y Levitin, On a new class of codes for identifying vertices in graphs. IEEE Trans. Inf. Theory 44, 599–611), es desafiante tanto desde el punto de vista teórico como computacional. En particular, se han propuesto algoritmos polinomiales para resolver el PCI sobre clases particulares de grafos y, más recientemente, se ha estudiado el poliedro asociado a su formulación natural donde, en algunos casos, se ha dado la descripción completa para algunas familias de grafos (véase el trabajo de Argiroffo, Bianchi y Wagler, Study of Identifying Code Polyhedra for Some Families of Split Graphs, LNCS 8596, 13–25). En esta comunicación reportamos algunos experimentos computacionales respecto a la performance de un modelo de programación entera para el PCI.Sociedad Argentina de Informática e Investigación Operativa (SADIO

Experimentos computacionales en la resolución del problema de códigos de identificación

El Problema de Códigos de Identificación (PCI) es un problema NP-difícil relativamente nuevo que, además de contar con aplicaciones concretas (véase el trabajo de Karpovsky, Chakrabarty y Levitin, On a new class of codes for identifying vertices in graphs. IEEE Trans. Inf. Theory 44, 599–611), es desafiante tanto desde el punto de vista teórico como computacional. En particular, se han propuesto algoritmos polinomiales para resolver el PCI sobre clases particulares de grafos y, más recientemente, se ha estudiado el poliedro asociado a su formulación natural donde, en algunos casos, se ha dado la descripción completa para algunas familias de grafos (véase el trabajo de Argiroffo, Bianchi y Wagler, Study of Identifying Code Polyhedra for Some Families of Split Graphs, LNCS 8596, 13–25). En esta comunicación reportamos algunos experimentos computacionales respecto a la performance de un modelo de programación entera para el PCI.Sociedad Argentina de Informática e Investigación Operativa (SADIO

On three domination numbers in block graphs

The problems of determining minimum identifying, locating-dominating or open locating-dominating codes are special search problems that are challenging both from a theoretical and a computational point of view. Hence, a typical line of attack for these problems is to determine lower and upper bounds for minimum codes in special graphs. In this work we study the problem of determining the cardinality of minimum codes in block graphs (that are diamond-free chordal graphs). We present for all three codes lower and upper bounds as well as block graphs where these bounds are attained

Coxeter group structure of cosmological billiards on compact spatial manifolds

We present a systematic study of the cosmological billiard structures of Einstein-p-form systems in which all spatial directions are compactified on a manifold of nontrivial topology. This is achieved for all maximally oxidised theories associated with split real forms, for all possible compactifications as defined by the de Rham cohomology of the internal manifold. In each case, we study the Coxeter group that controls the dynamics for energy scales below the Planck scale as well as the relevant billiard region. We compare and contrast them with the Weyl group and fundamental domain that emerge from the general BKL analysis. For generic topologies we find a variety of possibilities: (i) The group may or may not be a simplex Coxeter group; (ii) The billiard region may or may not be a fundamental domain. When it is not a fundamental domain, it can be described as a sequence of pairwise adjacent chambers, known as a gallery, and the reflections in the billiard walls provide a non-standard presentation of the Coxeter group. We find that it is only when the Coxeter group is a simplex Coxeter group, and the billiard region is a fundamental domain, that there is a correspondence between billiard walls and simple roots of a Kac-Moody algebra, as in the general BKL analysis. For each compactification we also determine whether or not the resulting theory exhibits chaotic dynamics.Comment: 51 pages. Typos corrected. References added. Submitted for publicatio

Minkowski Sum Construction and other Applications of Arrangements of Geodesic Arcs on the Sphere

We present two exact implementations of efficient output-sensitive algorithms that compute Minkowski sums of two convex polyhedra in 3D. We do not assume general position. Namely, we handle degenerate input, and produce exact results. We provide a tight bound on the exact maximum complexity of Minkowski sums of polytopes in 3D in terms of the number of facets of the summand polytopes. The algorithms employ variants of a data structure that represents arrangements embedded on two-dimensional parametric surfaces in 3D, and they make use of many operations applied to arrangements in these representations. We have developed software components that support the arrangement data-structure variants and the operations applied to them. These software components are generic, as they can be instantiated with any number type. However, our algorithms require only (exact) rational arithmetic. These software components together with exact rational-arithmetic enable a robust, efficient, and elegant implementation of the Minkowski-sum constructions and the related applications. These software components are provided through a package of the Computational Geometry Algorithm Library (CGAL) called Arrangement_on_surface_2. We also present exact implementations of other applications that exploit arrangements of arcs of great circles embedded on the sphere. We use them as basic blocks in an exact implementation of an efficient algorithm that partitions an assembly of polyhedra in 3D with two hands using infinite translations. This application distinctly shows the importance of exact computation, as imprecise computation might result with dismissal of valid partitioning-motions.Comment: A Ph.D. thesis carried out at the Tel-Aviv university. 134 pages long. The advisor was Prof. Dan Halperi

Two essays in computational optimization: computing the clar number in fullerene graphs and distributing the errors in iterative interior point methods

Fullerene are cage-like hollow carbon molecules graph of pseudospherical sym- metry consisting of only pentagons and hexagons faces. It has been the object of interest for chemists and mathematicians due to its widespread application in various fields, namely including electronic and optic engineering, medical sci- ence and biotechnology. A Fullerene molecular, Γ n of n atoms has a multiplicity of isomers which increases as N iso ∼ O(n 9 ). For instance, Γ 180 has 79,538,751 isomers. The Fries and Clar numbers are stability predictors of a Fullerene molecule. These number can be computed by solving a (possibly N P -hard) combinatorial optimization problem. We propose several ILP formulation of such a problem each yielding a solution algorithm that provides the exact value of the Fries and Clar numbers. We compare the performances of the algorithm derived from the proposed ILP formulations. One of this algorithm is used to find the Clar isomers, i.e., those for which the Clar number is maximum among all isomers having a given size. We repeated this computational experiment for all sizes up to 204 atoms. In the course of the study a total of 2 649 413 774 isomers were analyzed.The second essay concerns developing an iterative primal dual infeasible path following (PDIPF) interior point (IP) algorithm for separable convex quadratic minimum cost flow network problem. In each iteration of PDIPF algorithm, the main computational effort is solving the underlying Newton search direction system. We concentrated on finding the solution of the corresponding linear system iteratively and inexactly. We assumed that all the involved inequalities can be solved inexactly and to this purpose, we focused on different approaches for distributing the error generated by iterative linear solvers such that the convergences of the PDIPF algorithm are guaranteed. As a result, we achieved theoretical bases that open the path to further interesting practical investiga- tion