Maria J. Serna

A survey of graph layout problems

Graph layout problems are a particular class of combinatorial optimization problems whose goal is to find a linear layout of an input graph in such way that a certain objective cost is optimized. This survey considers their motivation, complexity, approximation properties, upper and lower bounds, heuristics and probabilistic analysis on random graphs. The result is a complete view of the current state of the art with respect to layout problems from an algorithmic point of view.

Randomization and Approximation Techniques in Computer Science

We present three algorithms to count the number of distinct elements in a data stream to within a factor of 1 ± . Our algorithms improve upon known algorithms for this problem, and offer a spectrum of time/space tradeoffs.

Approximating Layout Problems on Random Geometric Graphs

In this paper, we study the approximability of several layout problems on a family of random geometric graphs. Vertices of random geometric graphs are randomly distributed on the unit square and are connected by edges whenever they are closer than some given parameter. The layout problems that we consider are bandwidth, minimum linear arrangement, minimum cut width, minimum sum cut, vertex separation, and edge bisection. We first prove that some of these problems remain NP-complete even for geometric graphs. Afterwards, we compute lower bounds that hold, almost surely, for random geometric graphs. Then, we present two heuristics that, almost surely, turn out to be constant approximation algorithms for our layout problems on random geometric graphs. In fact, for the bandwidth and vertex separation problems, these heuristics are asymptotically optimal. Finally, we use the theoretical results in order to empirically compare these and other well-known heuristics.

A random graph model for optical networks of sensors

The main contribution of this paper is presenting a new model for Smart Dust networks communicating through optical links and showing its applicability when the goal of the network is monitoring an area under the surveillance of a base station. We analyze the basic parameters of these networks as a new model of random graphs and propose simple distributed protocols for basic communication. These protocols are designed to minimize the energy consumption.

Stability and non-stability of the FIFO protocol

In this paper, we analyze the stability properties of the FIFO protocol in the Adversarial Queueing model for packet routing. We show a graph for which FIFO is stable for any adversary with injection rate r ≰ 0.1428. We generalize this results to show upper bound for stability of any network under FIFO protocol, answering partially an open question raised by Andrews et al. in [2]. We also design a network and an adversary for which FIFO is non-stable for any r ≱ 0.8357, improving the previous known bounds of [2].

A Characterization of Universal Stability in the Adversarial Queuing Model

We study universal stability of directed and undirected graphs in the adversarial queuing model for static packet routing. In this setting, packets are injected in some edge and have to traverse a predefined path before leaving the system. Restrictions on the allowed packet trajectory provide a way to analyze stability under different packet trajectories. We consider five packet trajectories, two for directed graphs and three for undirected graphs, and provide polynomial time algorithms for testing universal stability when considering each of them. In each case we obtain a different characterization of the universal stability property in terms of a set of forbidden subgraphs. Thus we show that variations of the allowed packet trajectory lead to nonequivalent characterizations. Using those characterizations we are also able to provide polynomial time algorithms for testing stability under the \NTGLIS (Nearest To Go-Longest In System) protocol.

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 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 behaviour 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 behaviour of several layout measures on random geometric graphs in their subcritical regime. Our main results are convergence theorems that can be viewed as an analogue of the Beardwood, Halton and Hammersley theorem for the Euclidean TSP on random points in the unit square.

Random Geometric Problems on [0, 1]²

In this paper we survey the work done for graphs on random geometric models. We present some heuristics for the problem of the Minimal linear arrangement on [0,1]2 and we conclude with a collection of open problems.

On the complexity of metric dimension

The metric dimension of a graph G is the size of a smallest subset L⊆V(G) such that for any x,y∈V(G) there is a z∈L such that the graph distance between x and z differs from the graph distance between y and z. Even though this notion has been part of the literature for almost 40 years, the computational complexity of determining the metric dimension of a graph is still very unclear. Essentially, we only know the problem to be NP-hard for general graphs, to be polynomial-time solvable on trees, and to have a logn-approximation algorithm for general graphs. In this paper, we show tight complexity boundaries for the Metric Dimension problem. We achieve this by giving two complementary results. First, we show that the Metric Dimension problem on bounded-degree planar graphs is NP-complete. Then, we give a polynomial-time algorithm for determining the metric dimension of outerplanar graphs.

Approximating linear programming is log-space complete for P

Abstract We consider here two approximations of the general linear programming problem. A solution approximation requires a vector close to an optimal vector solution in some suitable norm. A value approximation seeks for a vector at which the objective function attains a value near to the optimum. We show that approximating within any factor ϵ > 0 any of those problems is P-complete under log-space reductions. In order to show the above result we prove the nonparallel approximability of computing the number of true gates in a Boolean circuit.