An optimal permutation routing algorithm on full-duplex hexagonal networks

Distributed Computing and NetworkingInternational audienceIn the permutation routing problem, each processor is the origin of at most one packet and the destination of no more than one packet. The goal is to minimize the number of time steps required to route all packets to their respective destinations, under the constraint that each link can be crossed simultaneously by no more than one packet. We study this problem in a hexagonal network, i.e. a finite subgraph of a triangular grid, which is a widely used network in practical applications. We present an optimal distributed permutation routing algorithm on full-duplex hexagonal networks, using the addressing scheme described by F.G. Nocetti, I. Stojmenovic and J. Zhang (IEEE TPDS 13(9): 962-971, 2002). Furthermore, we prove that this algorithm is oblivious and translation invariant

Edge-partitioning regular graphs for ring traffic grooming with a priori placement od the ADMs

We study the following graph partitioning problem: Given two positive integers C and Δ, find the least integer M(C,Δ) such that the edges of any graph with maximum degree at most Δ can be partitioned into subgraphs with at most C edges and each vertex appears in at most M(C,Δ) subgraphs. This problem is naturally motivated by traffic grooming, which is a major issue in optical networks. Namely, we introduce a new pseudodynamic model of traffic grooming in unidirectional rings, in which the aim is to design a network able to support any request graph with a given bounded degree. We show that optimizing the equipment cost under this model is essentially equivalent to determining the parameter M(C, Δ). We establish the value of M(C, Δ) for almost all values of C and Δ, leaving open only the case where Δ ≥ 5 is odd, Δ (mod 2C) is between 3 and C − 1, C ≥ 4, and the request graph does not contain a perfect matching. For these open cases, we provide upper bounds that differ from the optimal value by at most one.Peer ReviewedPostprint (published version

Parameterized Complexity of the Smallest Degree Constraint Subgraph Problem

In this paper we initiate the study of finding an induced subgraph of size at most $k$ with minimum degree at least $d$. We call this problem {\sc Minimum Subgraph of Minimum Degree $_{\geq d}$ (MSMD$_d$)}. For $d=2$, it corresponds to finding a shortest cycle of the graph. The problem is strongly related to the \textsc{Dense $k$-Subgraph} problem and is of interest in practical applications. We show that the {\sc MSMS}$_d$ is fixed parameter intractable for $d\geq 3$ in general graphs by showing it to be W[1]-hard by a reduction from {\sc Multi-Color Clique}. On the algorithmic side, we show that the problem is fixed parameter tractable in graphs which excluded minors and graphs with bounded local tree-width. In particular, this implies that the problem is fixed parameter tractable in planar graphs, graphs of bounded genus and graphs with bounded maximum degree

Hardness of Approximating the Traffic Grooming Problem

National audienceLe groupage est un problème central dans l'étude des réseaux optiques. Dans cet article, on propose le premier résultat d'inapproximabilité pour le problème du groupage, en affirmant la conjecture de Chow et Lin (2004, Networks, 44, 194-202), selon laquelle le groupage est APX-complet. On étudie aussi une version amortie du problème de sous-graphe le plus dense dans un graphe donné: trouver le sous-graphe de taille minimum ayant le degré minimum au moins d, d>=3. On démontre que ce dernier n'a pas d'approximation à un facteur constant

Drop cost and wavelength optimal two-period grooming with ratio 4

We study grooming for two-period optical networks, a variation of the traffic grooming problem for WDM ring networks introduced by Colbourn, Quattrocchi, and Syrotiuk. In the two-period grooming problem, during the first period of time, there is all-to-all uniform traffic among $n$ nodes, each request using $1/C$ of the bandwidth; and during the second period, there is all-to-all uniform traffic only among a subset $V$ of $v$ nodes, each request now being allowed to use $1/C'$ of the bandwidth, where $C' < C$. We determine the minimum drop cost (minimum number of ADMs) for any $n,v$ and C=4 and $C' \in \{1,2,3\}$. To do this, we use tools of graph decompositions. Indeed the two-period grooming problem corresponds to minimizing the total number of vertices in a partition of the edges of the complete graph $K_n$ into subgraphs, where each subgraph has at most $C$ edges and where furthermore it contains at most $C'$ edges of the complete graph on $v$ specified vertices. Subject to the condition that the two-period grooming has the least drop cost, the minimum number of wavelengths required is also determined in each case

GMPLS Label Space Minimization through Hypergraph Layouts

International audienceAll-Optical Label Switching (AOLS) is a new technology that performs packet forwarding without any optical-electrical-optical conversions. In this paper, we study the problem of routing a set of requests in AOLS networks using GMPLS technology, with the aim of minimizing the number of labels required to ensure the forwarding. We first formalize the problem by associating to each routing strategy a logical hypergraph, called a hypergraph layout, whose hyperarcs are dipaths of the physical graph, called tunnels in GMPLS terminology. We define a cost function for the hypergraph layout, depending on its total length plus its total hop count. Minimizing the cost of the design of an AOLS network can then be expressed as finding a minimum cost hypergraph layout. We prove hardness results for the problem, namely for general directed networks we prove that it is NP-hard to find a C log n-approximation, where C is a positive constant and n is the number of nodes of the network. For symmetric directed networks, we prove that the problem is APX-hard. These hardness results hold even if the traffic instance is a partial broadcast. On the other hand, we provide approximation algorithms, in particular an O(log n)-approximation for symmetric directed networks. Finally, we focus on the case where the physical network is a directed path, providing a polynomial-time dynamic programming algorithm for a fixed number k of sources running in O(n^{k+2}) time

Optimization in graphs under degree constraints. application to telecommunication networks

La premi ere partie de cette th ese s'int eresse au groupage de tra c dans les r eseaux de t el ecommunications. La notion de groupage de tra c correspond a l'agr egation de ux de faible d ebit dans des conduits de plus gros d ebit. Cependant, a chaque insertion ou extraction de tra c sur une longueur d'onde il faut placer dans le noeud du r eseau un multiplexeur a insertion/extraction (ADM). De plus il faut un ADM pour chaque longueur d'onde utilis ee dans le noeud, ce qui repr esente un co^ut d' equipements important. Les objectifs du groupage de tra c sont d'une part le partage e cace de la bande passante et d'autre part la r eduction du co^ut des equipements de routage. Nous pr esentons des r esultats d'inapproximabilit e, des algorithmes d'approximation, un nouveau mod ele qui permet au r eseau de pouvoir router n'importe quel graphe de requ^etes de degr e born e, ainsi que des solutions optimales pour deux sc enarios avec tra c all-to-all: l'anneau bidirectionnel et l'anneau unidirectionnel avec un facteur de groupage qui change de mani ere dynamique. La deuxi eme partie de la th ese s'interesse aux probl emes consistant a trouver des sousgraphes avec contraintes sur le degr e. Cette classe de probl emes est plus g en erale que le groupage de tra c, qui est un cas particulier. Il s'agit de trouver des sous-graphes d'un graphe donn e avec contraintes sur le degr e, tout en optimisant un param etre du graphe (tr es souvent, le nombre de sommets ou d'ar^etes). Nous pr esentons des algorithmes d'approximation, des resultats d'inapproximabilit e, des etudes sur la complexit e param etrique, des algorithmes exacts pour les graphes planaires, ainsi qu'une m ethodologie g en erale qui permet de r esoudre e cacement cette classe de probl emes (et de mani ere plus g en erale, la classe de probl emes tels qu'une solution peut ^etre cod e avec une partition d'un sous-ensemble des sommets) pour les graphes plong es dans une surface. Finalement, plurieurs annexes pr esentent des r esultats sur des probl emes connexes