Franco Robledo

Global Synchronization Properties for Different Classes of Underlying Interconnection Graphs for Kuramoto Coupled Oscillators

This article deals with the general ideas of almost global synchronization of Kuramoto coupled oscillators and synchronizing graphs. We review the main existing results and introduce new results for some classes of graphs that are important in network optimization: complete k -partite graphs and what we have called Monma graphs.

Network design with node connectivity constraints

The Generalized Steiner Problem with Node-Connectivity constraints (GSPNC) consists of determining of a minimum cost subnetwork of a given network where some pairs of nodes are required to satisfy node-connectivity requirements. The GSPNC has applications to the design of low-cost communications networks which can survive failures in the servers as well as in the connection lines. The GSPNC is known to be NP-Complete. In this paper, we introduce an algorithm based on GRASP (Greedy Randomized Adaptive Search Procedure), an effective combinatorial optimization metaheuristic. Experimental results are obtained over a set of problem instances with different characteristics and connectivity requirements, obtaining in all these cases optimal or near-optimal results.

Using GRASP for designing a layered network: a real IP/MPLS over DWDM application case

In this paper, we study a network design problem arising from the deployment of an IP/MPLS network over an existing transport infrastructure. The goal is to find a minimum cost installation of links such that traffic demands can resiliently be accomplished. In this paper, an integer programming formulation and metaheuristic are presented. This work is based on a real application case for a telecommunications company. The results allowed the company to reduce over 30% of the infrastructure costs.

The wheels: an infinite family of bi-connected planar synchronizing graphs

Almost global synchronization property of Kuramoto coupled oscillations was recently introduced and stands for the case where almost every initial condition of the dynamical system leads to the synchronization of all the agents. When the oscillators are all identical, the property only depends on the the underlying interconnection graph. If the property is present, the interconnection graph is called synchronizing. It is known that a graph is synchronizing if and only if its block are. So, the characterization of synchronizing graphs can be restricted to the class of bi-connected graphs. In this work, we present the first known infinite family of biconnected planar synchronizing graphs, named, the wheels. Besides, we present two graph wich are the first known chordal graph and Halin graphs that not synchronize.

Diameter-Constrained Reliability: Complexity, Factorization and Exact computation in Weak Graphs

In this paper we address a problem from the field of network reliability, called diameter-constrained reliability. Specifically, we are given a simple graph G = (V, E) with [V] = n nodes and [E] = m links, a subset K ⊆ V of terminals, a vector p = (p1,...,pm) &epsis; [0, 1]m and a positive integer d, called diameter. We assume nodes are perfect but links fail stochastically and independently, with probabilities qi = 1 --- pi. The diameter-constrained reliability (DCR for short), is the probability that the terminals of the resulting subgraph remain connected by paths composed by d links, or less. This number is denoted by RdK,G(p).n The general DCR computation is inside the class of NP-Hard problems, since is subsumes the complexity that a random graph is connected. In this paper the computational complexity of DCR-subproblems is discussed in terms of the number of terminal nodes k = [K] and diameter d. A factorization formula for exact DCR computation is provided, that runs in exponential time in the worst case. Finally, a revision of graph-classes that accept DCR computation in polynomial time is then included. In this class we have graphs with bounded co-rank, graphs with bounded genus, planar graphs, and, in particular, Monma graphs, which are relevant in robust network design. We extend this class adding arborescence graphs. A discussion of trends for future work is offered in the conclusions.

Optimal design of an IP/MPLS over DWDM network

Different approaches for deploying resilient optical networks of low cost constitute a traditional group of NP-Hard problems that have been widely studied. Most of them are based on the construction of low cost networks that fulfill connectivity constraints. However, recent trends to virtualize optical networks over the legacy fiber infrastructure, modified the nature of network design problems and turned inappropriate many of these models and algorithms. In this paper we study a design problem arising from the deployment of an IP/MPLS network over an existing DWDM infrastructure. Besides cost and resiliency, this problem integrates traffic and capacity constraints. We present: an integer programming formulation for the problem, theoretical results, and describe how several metaheuristics were applied in order to find good quality solutions, for a real application case of a telecommunications company.

Optimal Design of a Multi-Layer Network an IP/MPLS Over DWDM Application Case

In this paper we study a network design problem arising from the deployment of an IP/MPLS network over an existing transport infrastructure. The goal is to find a minimum cost installation of links such that traffic demands can resiliently be accomplished. We present an integer programming formulation for our problem and metaheuristics to find good quality solutions. This work is based on a real application case for a south american telecommunications company.

A cooperative network game efficiently solved via an ant colony optimization approach

In this paper, a Cooperative Network Game (CNG) is introduced. In this game, all players have the same goal: display a video content in real time, with no cuts and low buffering time. Inspired in cooperation and symmetry, all players should apply the same strategy, resulting in a fair play. For each strategy we shall define a score, and the search of the best one characterizes a Combinatorial Optimization Problem (COP). In this research we show that this search can be translated into a suitable Assymmetric Traveling Salesman Problem (ATSP). An Ant Colony Optimization (ACO) approach is defined, obtaining highly competitive solutions for the CNG. Finally, we play the game in a real context, using a new strategy in a Peer-to-Peer (P2P) platform, obtaining better results than previous strategies.

Graph Fragmentation Problem for Natural Disaster Management

Natural disasters represent a threaten for the existence of human beings. Given its remarkable importance, operational researchers should contribute to provide rationale decisions.

Analysis and complexity of pandemics.

Realistic epidemic models assume network propagation in a stochastic fashion, where the disease is disseminated through neighboring nodes. Here, we study node-immunization techniques, where the notion of immunization means node-deletion in a graph. In a highly virulent scenario, a pandemic takes effect, and the disease is spread all over the connected component of a graph. A combinatorial optimization problem is introduced, where the goal is to choose a node-immunization strategy to reduce the expected number of deaths in pandemics. We prove that this problem belongs to the NP-Complete class. As corollary, a large family of node-immunization problems arising from epidemic modelling are computationally hard as well. The value of the paper is to confirm the intuition behind the fact that it is hard to cope with epidemics. The paper is closed with heuristics in order to address the combinatorial problem for pandemic analysis.