Júlio Araújo

Energy efficient content distribution

To optimize energy efficiency in network, operators try to switch off as many network devices as possible. Recently, there is a trend to introduce content caches as an inherent capacity of network equipment, with the objective of improving the efficiency of content distribution and reducing network congestion. In this work, we study the impact of using in-network caches and content delivery network (CDN) cooperation on an energy-efficient routing. We formulate this problem as Energy Efficient Content Distribution. The objective is to find a feasible routing, so that the total energy consumption of the network is minimized subject to satisfying all the demands and link capacity. We exhibit the range of parameters (size of caches, popularity of content, demand intensity, etc.) for which caches are useful. Experimental results show that by placing a cache on each backbone router to store the most popular content, along with well choosing the best content provider server for each demand to a CDN, we can save a total up to 23% of power in the backbone, while 16% can be gained solely thanks to caches.

Hull number

In this paper, we study the (geodesic) hull number of graphs. For any two vertices u , v ź V of a connected undirected graph G = ( V , E ) , the closed interval I u , v of u and v is the set of vertices that belong to some shortest ( u , v ) -path. For any S ź V , let I S = ź u , v ź S I u , v . A subset S ź V is (geodesically) convex if I S = S . Given a subset S ź V , the convex hull I h S of S is the smallest convex set that contains S . We say that S is a hull set of G if I h S = V . The size of a minimum hull set of G is the hull number of G , denoted by h n ( G ) .First, we show a polynomial-time algorithm to compute the hull number of any P 5 -free triangle-free graph. Then, we present four reduction rules based on vertices with the same neighborhood. We use these reduction rules to propose a fixed parameter tractable algorithm to compute the hull number of any graphź G , where the parameter can be the size of a vertex cover of G or, more generally, its neighborhood diversity, and we also use these reductions to characterize the hull number of the lexicographic product of any two graphs.

Hybrid approaches for distributed storage systems

Distributed or peer-to-peer storage solutions rely on the introduction of redundant data to be fault-tolerant and to achieve high reliability. One way to introduce redundancy is by simple replication. This strategy allows an easy and fast access to data, and a good bandwidth efficiency to repair the missing redundancy when a peer leaves or fails in high churn systems. However, it is known that erasure codes, like Reed-Solomon, are an efficient solution in terms of storage space to obtain high durability when compared to replication. Recently, the Regenerating Codes were proposed as an improvement of erasure codes to better use the available bandwidth when reconstructing the missing information. In this work, we compare these codes with two hybrid approaches. The first was already proposed and mixes erasure codes and replication. The second one is a new proposal that we call Double Coding. We compare these approaches with the traditional Reed-Solomon code and also Regenerating Codes from the point of view of availability, durability and storage space. This comparison uses Markov Chain Models that take into account the reconstruction time of the systems.

Hull number: P5-free graphs and reduction rules

Abstract In this paper, we study the (geodesic) hull number of graphs. For any two vertices u , v ∈ V of a connected undirected graph G = ( V , E ) , the closed interval I [ u , v ] of u and v is the set of vertices that belong to some shortest ( u , v ) -path. For any S ⊆ V , let I [ S ] = ⋃ u , v ∈ S I [ u , v ] . A subset S ⊆ V is (geodesically) convex if I [ S ] = S . Given a subset S ⊆ V the convex hull I h [ S ] of S is the smallest convex set that contains S. We say that S is a hull set of G if I h [ S ] = V . The size of a minimum hull set of G is the hull number of G, denoted by h n ( G ) . First, we show a polynomial-time algorithm to compute the hull number of any P 5 -free triangle-free graph. Then, we present four reduction rules based on vertices with the same neighborhood. We use these reduction rules to propose a fixed parameter tractable algorithm to compute the hull number of any graph G, where the parameter can be the size of a vertex cover of G or, more generally, its neighborhood diversity, and we also use these reductions to characterize the hull number of the lexicographic product of any two graphs.

Grundy number on P4-classes

Abstract In this article, we define a new class of graphs, the fat-extended P 4 -laden graphs, and we show a polynomial time algorithm to determine the Grundy number of the graphs in this class. This result implies that the Grundy number can be found in polynomial time for any graph of the following classes: P4-reducible, extended P 4 -reducible, P 4 -sparse, extended P 4 -sparse, P 4 -extendible, P 4 -lite, P 4 -tidy, P 4 -laden and extended P 4 -laden, which are all strictly contained in the fat-extended P 4 -laden class.

On the proper orientation number of bipartite graphs

An orientation of a graph G is a digraph D obtained from G by replacing each edge by exactly one of the two possible arcs with the same endvertices. For each v ? V ( G ) , the indegree of v in D, denoted by d D - ( v ) , is the number of arcs with head v in D. An orientation D of G is proper if d D - ( u ) ? d D - ( v ) , for all u v ? E ( G ) . The proper orientation number of a graph G, denoted by ? ? ( G ) , is the minimum of the maximum indegree over all its proper orientations. In this paper, we prove that ? ? ( G ) ? ( Δ ( G ) + Δ ( G ) ) / 2 + 1 if G is a bipartite graph, and ? ? ( G ) ? 4 if G is a tree. It is well-known that ? ? ( G ) ? Δ ( G ) , for every graph G. However, we prove that deciding whether ? ? ( G ) ? Δ ( G ) - 1 is already an NP -complete problem on graphs with Δ ( G ) = k , for every k ? 3 . We also show that it is NP -complete to decide whether ? ? ( G ) ? 2 , for planar subcubic graphs G. Moreover, we prove that it is NP -complete to decide whether ? ? ( G ) ? 3 , for planar bipartite graphs G with maximum degree 5.

Eulerian and Hamiltonian dicycles in directed hypergraphs

In this paper, we generalize the concepts of Eulerian and Hamiltonian digraphs to directed hypergraphs. A dihypergraphH is a pair (𝒱(H), ℰ(H)), where 𝒱(H) is a non-empty set of elements, called vertices, and ℰ(H) is a collection of ordered pairs of subsets of 𝒱(H), called hyperarcs. It is Eulerian (resp. Hamiltonian) if there is a dicycle containing each hyperarc (resp. each vertex) exactly once. We first present some properties of Eulerian and Hamiltonian dihypergraphs. For example, we show that deciding whether a dihypergraph is Eulerian is an NP-complete problem. We also study when iterated line dihypergraphs are Eulerian and Hamiltonian. Finally, we study when the generalized de Bruijn dihypergraphs are Eulerian and Hamiltonian. In particular, we determine when they contain a complete Berge dicycle, i.e., an Eulerian and Hamiltonian dicycle.

Connectivity Inference in Mass Spectrometry Based Structure Determination

We consider the following Minimum Connectivity Inference problem (MCI), which arises in structural biology: given vertex sets V i ⊆ V, i ∈ I, find a graph G = (V,E) minimizing the size of the edge set E, such that the sub-graph of G induced by each V i is connected. This problem arises in structural biology, when one aims at finding the pairwise contacts between the proteins of a protein assembly, given the lists of proteins involved in sub-complexes. We present four contributions.

Weighted improper colouring

In this paper, we study a colouring problem motivated by a practical frequency assignment problem and up to our best knowledge new. In wireless networks, a node interferes with the other nodes the level of interference depending on numerous parameters: distance between the nodes, geographical topography, obstacles, etc. We model this with a weighted graph G where the weights on the edges represent the noise (interference) between the two end-nodes. The total interference in a node is then the sum of all the noises of the nodes emitting on the same frequency. A weighted t-improper k-colouring of G is a k-colouring of the nodes of G (assignment of k frequencies) such that the interference at each node does not exceed some threshold t. The Weighted Improper Colouring problem, that we consider here consists in determining the weighted t-improper chromatic number defined as the minimum integer k such that G admits a weighted t-improper k-colouring. We also consider the dual problem, denoted the Threshold Improper Colouring problem, where given a number k of colours (frequencies) we want to determine the minimum real t such that G admits a weighted t-improper k-colouring. We show that both problems are NP-hard and first present general upper bounds; in particular we show a generalisation of Lovaszs Theorem for the weighted t-improper chromatic number. We then show how to transform an instance of the Threshold Improper Colouring problem into another equivalent one where the weights are either 1 or M, for a sufficient big value M. Motivated by the original application, we study a special interference model on various grids (square, triangular, hexagonal) where a node produces a noise of intensity 1 for its neighbours and a noise of intensity 1/2 for the nodes that are at distance 2. Consequently, the problem consists of determining the weighted t-improper chromatic number when G is the square of a grid and the weights of the edges are 1, if their end nodes are adjacent in the grid, and 1/2 otherwise. Finally, we model the problem using linear integer programming, propose and test heuristic and exact Branch-and-Bound algorithms on random cell-like graphs, namely the Poisson-Voronoi tessellations.

Energy Efficient Content Distribution

To optimize energy efficiency in network, operators try to switch off as many network devices as possible. Recently, there is a trend to introduce content caches as an inherent capacity of network equipment, with the objective of improving the efficiency of content distribution and reducing network congestion. In this work, we study the impact of using in-network caches and content delivery network (CDN) cooperation on an energy-efficient routing. We formulate this problem as Energy Efficient Content Distribution. The objective is to find a feasible routing, so that the total energy consumption of the network is minimized subject to satisfying all the demands and link capacity. We exhibit the range of parameters (size of caches, popularity of content, demand intensity, etc.) for which caches are useful. Experimental results show that by placing a cache on each backbone router to store the most popular content, along with well choosing the best content provider server for each demand to a CDN, we can save a total up to 23% of power in the backbone, while 16% can be gained solely thanks to caches.