Results 1  10
of
13
Graph Layout Problems Parameterized by Vertex Cover
"... In the framework of parameterized complexity, one of the most commonly used structural parameters is the treewidth of the input graph. The reason for this is that most natural graph problems turn out to be fixed parameter tractable when parameterized by treewidth. However, Graph Layout problems are ..."
Abstract

Cited by 9 (3 self)
 Add to MetaCart
In the framework of parameterized complexity, one of the most commonly used structural parameters is the treewidth of the input graph. The reason for this is that most natural graph problems turn out to be fixed parameter tractable when parameterized by treewidth. However, Graph Layout problems are a notable exception. In particular, no fixed parameter tractable algorithms are known for the Cutwidth, Bandwidth, Imbalance and Distortion problems parameterized by treewidth. In fact, Bandwidth remains NPcomplete even restricted to trees. A possible way to attack graph layout problems is to consider structural parameterizations that are stronger than treewidth. In this paper we study graph layout problems parameterized by the size of the minimum vertex cover of the input graph. We show that all the mentioned problems are fixed parameter tractable. Our basic ingredient is a classical algorithm for Integer Linear Programming when parameterized by dimension, designed by Lenstra and later improved by Kannan. We hope that our results will serve to reemphasize the importance and utility of this algorithm.
A Polynomial Time Algorithm for the cutwidth of bounded degree graphs with small treewidth
, 2001
"... The cutwidth of a graph G is defined to be the smallest integer k such that the vertices of G can be arranged in a vertex ordering... ..."
Abstract

Cited by 6 (2 self)
 Add to MetaCart
The cutwidth of a graph G is defined to be the smallest integer k such that the vertices of G can be arranged in a vertex ordering...
Convergence Theorems for Some Layout Measures on Random Lattice and Random Geometric Graphs
"... This work deals with convergence theorems and bounds on the cost of several layout measures for lattice graphs, random lattice graphs and sparse random geometric graphs. Specifically, we consider the following problems: Minimum Linear Arrangement, Cutwidth, Sum Cut, Vertex Separation, Edge Bisection ..."
Abstract

Cited by 5 (0 self)
 Add to MetaCart
This work deals with convergence theorems and bounds on the cost of several layout measures for lattice graphs, random lattice graphs and sparse random geometric graphs. Specifically, we consider the following problems: Minimum Linear Arrangement, Cutwidth, Sum Cut, Vertex Separation, Edge Bisection and Vertex Bisection. For full square lattices, we give optimal layouts for the problems still open. For arbitrary lattice graphs, we present best possible bounds disregarding a constant factor. We apply percolation theory to the study of lattice graphs in a probabilistic setting. In particular, we deal with the subcritical regime that this class of graphs exhibits and characterize the behavior of several layout measures in this space of probability. We extend the results on random lattice graphs to random geometric graphs, which are graphs whose nodes are spread at random in the unit square and whose edges connect pairs of points which are within a given distance. We also characterize the behavior of several layout measures on random geometric graphs in their subcritical regime. Our main results are convergence theorems that can be viewed as an analog of the Beardwood, Halton and Hammersley theorem for the Euclidean TSP on random points in the unit square.
Practical Performance of Efficient Minimum Cut Algorithms
, 1997
"... In the early nineties, three major exciting new developments (and some ramifications) in the computation of minimum capacity cuts occurred and these developments motivated us to evaluate the old and new methods experimentally. We provide a brief overview of the most important algorithms for the mini ..."
Abstract

Cited by 4 (1 self)
 Add to MetaCart
In the early nineties, three major exciting new developments (and some ramifications) in the computation of minimum capacity cuts occurred and these developments motivated us to evaluate the old and new methods experimentally. We provide a brief overview of the most important algorithms for the minimum cut problem and compare these methods both on problem instances from the literature and graphs originating from the solution of the traveling salesman problem by branchandcut. 1. Introduction Computer programs that compute minimum capacity cuts in undirected graphs with nonnegative edge capacities belong to the most intensively used basic tools in optimization. Besides the obvious direct application of deciding the degree of connectivity of a given network, the main reason is that various separation routines in cutting plane algorithms depend on the practically efficient computation of minimum capacity cuts. The most prominent example is the separation of subtour elimination constrain...
Graph Drawing Algorithm Engineering with AGD
, 2000
"... We discuss the algorithm engineering aspects of AGD, a software library of algorithms for graph drawing. AGD represents algorithms as classes that provide one or more methods for calling the algorithm. There is a common base class, also called the type of an algorithm, for algorithms providing basic ..."
Abstract

Cited by 3 (2 self)
 Add to MetaCart
We discuss the algorithm engineering aspects of AGD, a software library of algorithms for graph drawing. AGD represents algorithms as classes that provide one or more methods for calling the algorithm. There is a common base class, also called the type of an algorithm, for algorithms providing basically the same functionality. This enables us to exchange components and experiment with various algorithms and implementations of the same type. We give examples for algorithm engineering with AGD for drawing general nonhierarchical graphs and hierarchical graphs.
BoundaryOptimal Triangulation Flooding
"... Given a planar triangulation all of whose faces are initially white, we study the problem of colouring the faces black one by one so that the boundary between black and white faces as well as the number of connected black and white regions are small at all times. We call such a colouring sequence o ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
Given a planar triangulation all of whose faces are initially white, we study the problem of colouring the faces black one by one so that the boundary between black and white faces as well as the number of connected black and white regions are small at all times. We call such a colouring sequence of the triangles a flooding. Our main result shows that it is in general impossible to guarantee boundary size O(n 1−ɛ), for any ɛ> 0, and a number of regions that is o(log n), where n is the number of faces of the triangulation. We also show that a flooding with boundary size O (√n) and O(log n) regions can be computed in O(n log n) time.