Solving the Canonical Representation and Star System Problems for Proper Circular-Arc Graphs in Log-Space

We present a logspace algorithm that constructs a canonical intersection model for a given proper circular-arc graph, where `canonical' means that models of isomorphic graphs are equal. This implies that the recognition and the isomorphism problems for this class of graphs are solvable in logspace. For a broader class of concave-round graphs, that still possess (not necessarily proper) circular-arc models, we show that those can also be constructed canonically in logspace. As a building block for these results, we show how to compute canonical models of circular-arc hypergraphs in logspace, which are also known as matrices with the circular-ones property. Finally, we consider the search version of the Star System Problem that consists in reconstructing a graph from its closed neighborhood hypergraph. We solve it in logspace for the classes of proper circular-arc, concave-round, and co-convex graphs.Comment: 19 pages, 3 figures, major revisio

Solving Vertex Cover in Polynomial Time on Hyperbolic Random Graphs

The VertexCover problem is proven to be computationally hard in different ways: It is NP-complete to find an optimal solution and even NP-hard to find an approximation with reasonable factors. In contrast, recent experiments suggest that on many real-world networks the run time to solve VertexCover is way smaller than even the best known FPT-approaches can explain. Similarly, greedy algorithms deliver very good approximations to the optimal solution in practice. We link these observations to two properties that are observed in many real-world networks, namely a heterogeneous degree distribution and high clustering. To formalize these properties and explain the observed behavior, we analyze how a branch-and-reduce algorithm performs on hyperbolic random graphs, which have become increasingly popular for modeling real-world networks. In fact, we are able to show that the VertexCover problem on hyperbolic random graphs can be solved in polynomial time, with high probability. The proof relies on interesting structural properties of hyperbolic random graphs. Since these predictions of the model are interesting in their own right, we conducted experiments on real-world networks showing that these properties are also observed in practice. When utilizing the same structural properties in an adaptive greedy algorithm, further experiments suggest that, on real instances, this leads to better approximations than the standard greedy approach within reasonable time

Two problems in computational geometry

En aquesta tesi s'estudien dos problemes del camp de la geometria computacional. El primer problema és: donat un set S de n punts en el pla en posició general, com de prop són quatre punts de S de ser cocirculars. Definim tres mesures per estudiar aquesta qüestió, la mesura de Tales, la mesura de Voronoi, i la mesura del Determinant. Presentem cotes per la mesura de Tales, i algoritmes per computar aquestes mesures de cocircularitat. També reduïm el problema de computar la cocircularitat emprant la mesura del Determinant al problema de 4SUM. El segon problema és: donat dos sets R i B de punts rojos i blaus respectivament, com computar la discrepància bicromàtica amb caixes i cercles. La discrepància bicromàtica és definida com la diferència entre el nombre de punts vermells i blaus que són a l'interior de la figura examinada. Presentem una comparativa entre algoritmes ja existents per les dues figures. També comparem la discrepància bicromàtica de caixes orientades en els eixos vs. d'orientació general. A més a més, també presentem un nou algoritme per la discrepància en esferes/discs per a altes dimensions, basat en literatura ja existent. També relacionem altres problemes en el tema de separabilitat amb algoritmes sensitius a l'output per la discrepància amb caixes.En esta tesis se estudian dos problemas del campo de la geometría computacional. El primer problema es: dado un set S de n puntos en el plan en posición general, como de cerca son cuatro puntos de S de ser cocirculares. Definimos tres medidas para estudiar esta cuestión, la medida de Tales, la medida de Voronoi, y la medida del Determinante. Presentamos cotas por la medida de Tales, y algoritmos para computar estas medidas de cocircularidad. También reducimos el problema de computar la cocircularidad usando la medida del Determinante al problema de 4SUM. El segundo problema es: dado dos sets R y B de puntos rojos y azules respectivamente, como computar la discrepancia bicromática con cajas y círculos. La discrepancia bicromática es definida como la diferencia entre el número de puntos rojos y azules que están en el interior de la figura examinada. Presentamos una comparativa entre algoritmos ya existentes por las dos figuras. También comparamos la discrepancia bicromática de cajas orientadas en los ejes vs. de orientación general. Además, también presentamos un nuevo algoritmo por la discrepancia en esferas/discos para altas dimensiones, basado en literatura ya existente. También relacionamos otros problemas en el tema de separabilidad con algoritmos sensitivos al output por la discrepancia con cajas.Two different problems belonging to computational geometry are studied in this thesis. The first problem studies: given a set S of n points in the plane in general position, how close are four points of S to being cocircular. We define three measures to study this question, the Thales, Voronoi and Determinant measures. We present bounds on the Thales almost-cocircularity measure over a point set. Algorithms for computing these measures of cocircularity are presented as well. We give a reduction from computing cocircularity using the Determinant measure to the 4SUM problem. The second problem studies: given two sets R and B of red and blue points respectively, how to compute the bichromatic discrepancy using boxes and circles. The bichromatic discrepancy is defined as the difference between the number of red points and blue points inside the shape. We present a comparison of algorithms in the existing literature for the two shapes. Bichromatic discrepancy in axis-parallel boxes .vs non-axis-parallel boxes is also compared. Furthermore, we also present a new algorithm for disk discrepancy in higher dimensions, based on existing literature. We also relate existing problems in separability with existing output sensitive algorithms for bichromatic discrepancy using boxes

Solving Vertex Cover in Polynomial Time on Hyperbolic Random Graphs

The computational complexity of the VERTEXCOVER problem has been studied extensively. Most notably, it is NP-complete to find an optimal solution and typically NP-hard to find an approximation with reasonable factors. In contrast, recent experiments suggest that on many real-world networks the run time to solve VERTEXCOVER is way smaller than even the best known FPT-approaches can explain. We link these observations to two properties that are observed in many real-world networks, namely a heterogeneous degree distribution and high clustering. To formalize these properties and explain the observed behavior, we analyze how a branch-and-reduce algorithm performs on hyperbolic random graphs, which have become increasingly popular for modeling real-world networks. In fact, we are able to show that the VERTEXCOVER problem on hyperbolic random graphs can be solved in polynomial time, with high probability. The proof relies on interesting structural properties of hyperbolic random graphs. Since these predictions of the model are interesting in their own right, we conducted experiments on real-world networks showing that these properties are also observed in practice

Verifying black hole orbits with gravitational spectroscopy

Gravitational waves from test masses bound to geodesic orbits of rotating black holes are simulated, using Teukolsky's black hole perturbation formalism, for about ten thousand generic orbital configurations. Each binary radiates power exclusively in modes with frequencies that are integer-linear-combinations of the orbit's three fundamental frequencies. The following general spectral properties are found with a survey of orbits: (i) 99% of the radiated power is typically carried by a few hundred modes, and at most by about a thousand modes, (ii) the dominant frequencies can be grouped into a small number of families defined by fixing two of the three integer frequency multipliers, and (iii) the specifics of these trends can be qualitatively inferred from the geometry of the orbit under consideration. Detections using triperiodic analytic templates modeled on these general properties would constitute a verification of radiation from an adiabatic sequence of black hole orbits and would recover the evolution of the fundamental orbital frequencies. In an analogy with ordinary spectroscopy, this would compare to observing the Bohr model's atomic hydrogen spectrum without being able to rule out alternative atomic theories or nuclei. The suitability of such a detection technique is demonstrated using snapshots computed at 12-hour intervals throughout the last three years before merger of a kludged inspiral. Because of circularization, the number of excited modes decreases as the binary evolves. A hypothetical detection algorithm that tracks mode families dominating the first 12 hours of the inspiral would capture 98% of the total power over the remaining three years.Comment: 18 pages, expanded section on detection algorithms and made minor edits. Final published versio