Josep Díaz

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.

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.

Algorithms for Learning Finite Automata from Queries: A Unified View

In this survey we compare several known variants of the algorithm for learning deterministic finite automata via membership and equivalence queries. We believe that our presentation makes it easier to understand what is going on and what the differences between the various algorithms mean. We also include the comparative analysis of the algorithms, review some known lower bounds, prove a new one, and discuss the question of parallelizing this sort of algorithm.

Classes of bounded nondeterminism

We study certain language classes located betweenP andNP that are defined by polynomial-time machines with a bounded amount of nondeterminism. We observe that these classes have complete problems and find a characterization of the classes using robust machines with bounded access to the oracle, obtaining some other results in this direction. We also study questions related to the existence of complete tally sets in these classes and closure of the classes under different types of polynomial-time reducibilities.

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].

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.

On the satisfiability threshold of formulas with three literals per clause

In this paper we present a new upper bound for randomly chosen 3-CNF formulas. In particular we show that any random formula over n variables, with a clauses-to-variables ratio of at least 4.4898 is, as n grows large, asymptotically almost surely unsatisfiable. The previous best such bound, due to Dubois in 1999, was 4.506. The first such bound, independently discovered by many groups of researchers since 1983, was 5.19. Several decreasing values between 5.19 and 4.506 were published in the years between. We believe that the probabilistic techniques we use for the proof are of independent interest.