Below All Subsets for Some Permutational Counting Problems

We show that the two problems of computing the permanent of an $n\times n$ matrix of $\operatorname{poly}(n)$-bit integers and counting the number of Hamiltonian cycles in a directed $n$-vertex multigraph with $\operatorname{exp}(\operatorname{poly}(n))$ edges can be reduced to relatively few smaller instances of themselves. In effect we derive the first deterministic algorithms for these two problems that run in $o(2^n)$ time in the worst case. Classic $\operatorname{poly}(n)2^n$ time algorithms for the two problems have been known since the early 1960's. Our algorithms run in $2^{n-\Omega(\sqrt{n/\log n})}$ time.Comment: Corrected several technical errors, added comment on how to use the algorithm for ATSP, and changed title slightly to a more adequate on

On connectedness and hamiltonicity of direct graph bundles

A necessary and sufficient condition for connectedness of direct graph bundles is given where the fibers are cycles. We also prove that all connected direct graph bundles $X=C_stimes^{alpha}C_t$ are Hamiltonian

Pseudo-random graphs

Random graphs have proven to be one of the most important and fruitful concepts in modern Combinatorics and Theoretical Computer Science. Besides being a fascinating study subject for their own sake, they serve as essential instruments in proving an enormous number of combinatorial statements, making their role quite hard to overestimate. Their tremendous success serves as a natural motivation for the following very general and deep informal questions: what are the essential properties of random graphs? How can one tell when a given graph behaves like a random graph? How to create deterministically graphs that look random-like? This leads us to a concept of pseudo-random graphs and the aim of this survey is to provide a systematic treatment of this concept.Comment: 50 page

Two Results in Drawing Graphs on Surfaces

In this work we present results on crossing-critical graphs drawn on non-planar surfaces and results on edge-hamiltonicity of graphs on the Klein bottle. We first give an infinite family of graphs that are 2-crossing-critical on the projective plane. Using this result, we construct 2-crossing-critical graphs for each non-orientable surface. Next, we use 2-amalgamations to construct 2-crossing-critical graphs for each orientable surface other than the sphere. Finally, we contribute to the pursuit of characterizing 4-connected graphs that embed on the Klein bottle and fail to be edge-hamiltonian. We show that known 4-connected counterexamples to edge-hamiltonicity on the Klein bottle are hamiltonian and their structure allows restoration of edge-hamiltonicity with only a small change

Below All Subsets for Some Permutational Counting Problems

We show that the two problems of computing the permanent of an n*n matrix of poly(n)-bit integers and counting the number of Hamiltonian cycles in a directed n-vertex multigraph with exp(poly(n)) edges can be reduced to relatively few smaller instances of themselves. In effect we derive the first deterministic algorithms for these two problems that run in o(2^n) time in the worst case. Classic poly(n)2^n time algorithms for the two problems have been known since the early 1960\u27s. Our algorithms run in 2^{n-Omega(sqrt{n/log(n)})} time

Spotting Trees with Few Leaves

We show two results related to the Hamiltonicity and $k$-Path algorithms in undirected graphs by Bj\"orklund [FOCS'10], and Bj\"orklund et al., [arXiv'10]. First, we demonstrate that the technique used can be generalized to finding some $k$-vertex tree with $l$ leaves in an $n$-vertex undirected graph in $O^*(1.657^k2^{l/2})$ time. It can be applied as a subroutine to solve the $k$-Internal Spanning Tree ($k$-IST) problem in $O^*(\min(3.455^k, 1.946^n))$ time using polynomial space, improving upon previous algorithms for this problem. In particular, for the first time we break the natural barrier of $O^*(2^n)$. Second, we show that the iterated random bipartition employed by the algorithm can be improved whenever the host graph admits a vertex coloring with few colors; it can be an ordinary proper vertex coloring, a fractional vertex coloring, or a vector coloring. In effect, we show improved bounds for $k$-Path and Hamiltonicity in any graph of maximum degree $\Delta=4,\ldots,12$ or with vector chromatic number at most 8

Generalized Delaunay triangulations : graph-theoretic properties and algorithms

This thesis studies different generalizations of Delaunay triangulations, both from a combinatorial and algorithmic point of view. The Delaunay triangulation of a point set S, denoted DT(S), has vertex set S. An edge uv is in DT(S) if it satisfies the empty circle property: there exists a circle with u and v on its boundary that does not enclose points of S. Due to different optimization criteria, many generalizations of the DT(S) have been proposed. Several properties are known for DT(S), yet, few are known for its generalizations. The main question we explore is: to what extent can properties of DT(S) be extended for generalized Delaunay graphs? First, we explore the connectivity of the flip graph of higher order Delaunay triangulations of a point set S in the plane. The order-k flip graph might be disconnected for k = 3, yet, we give upper and lower bounds on the flip distance from one order-k triangulation to another in certain settings. Later, we show that there exists a length-decreasing sequence of plane spanning trees of S that converges to the minimum spanning tree of S with respect to an arbitrary convex distance function. Each pair of consecutive trees in the sequence is contained in a constrained convex shape Delaunay graph. In addition, we give a linear upper bound and specific bounds when the convex shape is a square. With focus still on convex distance functions, we study Hamiltonicity in k-order convex shape Delaunay graphs. Depending on the convex shape, we provide several upper bounds for the minimum k for which the k-order convex shape Delaunay graph is always Hamiltonian. In addition, we provide lower bounds when the convex shape is in a set of certain regular polygons. Finally, we revisit an affine invariant triangulation, which is a special type of convex shape Delaunay triangulation. We show that many properties of the standard Delaunay triangulations carry over to these triangulations. Also, motivated by this affine invariant triangulation, we study different triangulation methods for producing other affine invariant geometric objects.Esta tesis estudia diferentes generalizaciones de la triangulación de Delaunay, tanto desde un punto de vista combinatorio como algorítmico. La triangulación de Delaunay de un conjunto de puntos S, denotada DT(S), tiene como conjunto de vértices a S. Una arista uv está en DT(S) si satisface la propiedad del círculo vacío: existe un círculo con u y v en su frontera que no contiene ningún punto de S en su interior. Debido a distintos criterios de optimización, se han propuesto varias generalizaciones de la DT (S). Hoy en día, se conocen bastantes propiedades de la DT(S), sin embargo, poco se sabe sobre sus generalizaciones. La pregunta principal que exploramos es: ¿Hasta qué punto las propiedades de la DT(S) se pueden extender para generalizaciones de gráficas de Delaunay? Primero, exploramos la conectividad de la gráfica de flips de las triangulaciones de Delaunay de orden alto de un conjunto de puntos S en el plano. La gráfica de flips de triangulaciones de orden k = 3 podría ser disconexa, sin embargo, nosotros damos una cota superior e inferior para la distancia en flips de una triangulación de orden k a alguna otra cuando S cumple con ciertas características. Luego, probamos que existe una secuencia de árboles generadores sin cruces tal que la suma total de la longitud de las aristas con respecto a una distancia convexa arbitraria es decreciente y converge al árbol generador mínimo con respecto a la distancia correspondiente. Cada par de árboles consecutivos en la secuencia se encuentran en una triangulación de Delaunay con restricciones. Adicionalmente, damos una cota superior lineal para la longitud de la secuencia y cotas específicas cuando el conjunto convexo es un cuadrado. Aún concentrados en distancias convexas, estudiamos hamiltonicidad en las gráficas de Delaunay de distancia convexa de k-orden. Dependiendo en la distancia convexa, exhibimos diversas cotas superiores para el mínimo valor de k que satisface que la gráfica de Delaunay de distancia convexa de orden-k es hamiltoniana. También damos cotas inferiores para k cuando el conjunto convexo pertenece a un conjunto de ciertos polígonos regulares. Finalmente, re-visitamos una triangulación afín invariante, la cual es un caso especial de triangulación de Delaunay de distancia convexa. Probamos que muchas propiedades de la triangulación de Delaunay estándar se preservan en estas triangulaciones. Además, motivados por esta triangulación afín invariante, estudiamos diferentes algoritmos que producen otros objetos geométricos afín invariantes