Placing regenerators in optical networks to satisfy multiple sets of requests.

The placement of regenerators in optical networks has become an active area of research during the last years. Given a set of lightpaths in a network G and a positive integer d, regenerators must be placed in such a way that in any lightpath there are no more than d hops without meeting a regenerator. While most of the research has focused on heuristics and simulations, the first theoretical study of the problem has been recently provided in [10], where the considered cost function is the number of locations in the network hosting regenerators. Nevertheless, in many situations a more accurate estimation of the real cost of the network is given by the total number of regenerators placed at the nodes, and this is the cost function we consider. Furthermore, in our model we assume that we are given a finite set of p possible traffic patterns (each given by a set of lightpaths), and our objective is to place the minimum number of regenerators at the nodes so that each of the traffic patterns is satisfied. While this problem can be easily solved when d = 1 or p = 1, we prove that for any fixed d,p ≥ 2 it does not admit a PTASUnknown control sequence '\textsc', even if G has maximum degree at most 3 and the lightpaths have length O(d)(d). We complement this hardness result with a constant-factor approximation algorithm with ratio ln (d ·p). We then study the case where G is a path, proving that the problem is NP-hard for any d,p ≥ 2, even if there are two edges of the path such that any lightpath uses at least one of them. Interestingly, we show that the problem is polynomial-time solvable in paths when all the lightpaths share the first edge of the path, as well as when the number of lightpaths sharing an edge is bounded. Finally, we generalize our model in two natural directions, which allows us to capture the model of [10] as a particular case, and we settle some questions that were left open in [10]

Multicasting in All-Optical WDM Networks

n this dissertation, we study the problem of (i) routing and wavelength assignment, and (ii) traffic grooming for multicast traffic in Wavelength Division Multiplexing (WDM) based all-optical networks. We focus on the 'static' case where the set of multicast traffic requests is assumed to be known in advance. For the routing and wavelength assignment problem, we study the objective of minimizing the number of wavelengths required; and for the traffic grooming problem, we study the objectives of minimizing (i) the number of wavelengths required, and (ii) the number of electronic components required. Both the problems are known to be hard for general fiber network topologies. Hence, it makes sense to study the problems under some restrictions on the network topology. We study the routing and wavelength assignment problem for bidirected trees, and the traffic grooming problem for unidirectional rings. The selected topologies are simple in the sense that the routing for any multicast traffic request is trivially determined, yet complex in the sense that the overall problems still remain hard. A motivation for selecting these topologies is that they are of practical interest since most of the deployed optical networks can be decomposed into these elemental topologies. In the first part of the thesis, we study the the problem of multicast routing and wavelength assignment in all-optical bidirected trees with the objective of minimizing the number of wavelengths required in the network. We give a 5/2-approximation algorithm for the case when the degree of the bidirected tree is at most 3. We give another algorithm with approximation ratio 10/3, 3 and 2 for the case when the degree of the bidirected tree is equal to 4, 3 and 2, respectively. The time complexity analysis for both these algorithms is also presented. Next we prove that the problem is hard even for the two restricted cases when the bidirected tree has (i) depth 2, and (ii) degree 2. Finally, we present another hardness result for a related problem of finding the clique number for a class for intersection graphs. In the second part of the thesis, we study the problem of multicast traffic grooming in all-optical unidirectional rings. For the case when the objective is to minimize the number of wavelengths required in the network, given an 'a'-approximation algorithm for the circular arc coloring problem, we give an algorithm having asymptotic approximation ratio 'a' for the multicast traffic grooming problem. We develop an easy to calculate lower bound on the minimum number of electronic components required to support a given set of multicast traffic requests on a given unidirectional ring network. We use this lower bound to analyze the worst case performance of a pair of simple grooming schemes. We also study the case when no grooming is carried out in order to get an estimate on the maximum number of electronic components that can be saved by applying intelligent grooming. Finally, we present a new grooming scheme and compare its average performance against other grooming schemes via simulations. The time complexity analysis for all the grooming schemes is also presented

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

Traffic Grooming in Bidirectional WDM Ring Networks

We study the minimization of ADMs (Add-Drop Multiplexers) in optical WDM bidirectional rings considering symmetric shortest path routing and all-to-all unitary requests. We precisely formulate the problem in terms of graph decompositions, and state a general lower bound for all the values of the grooming factor $C$ and $N$, the size of the ring. We first study exhaustively the cases $C=1$, $C = 2$, and $C=3$, providing improved lower bounds, optimal constructions for several infinite families, as well as asymptotically optimal constructions and approximations. We then study the case $C>3$, focusing specifically on the case $C = k(k+1)/2$ for some $k \geq 1$. We give optimal decompositions for several congruence classes of $N$ using the existence of some combinatorial designs. We conclude with a comparison of the cost functions in unidirectional and bidirectional WDM rings

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

Path computation in multi-layer networks: Complexity and algorithms

Carrier-grade networks comprise several layers where different protocols coexist. Nowadays, most of these networks have different control planes to manage routing on different layers, leading to a suboptimal use of the network resources and additional operational costs. However, some routers are able to encapsulate, decapsulate and convert protocols and act as a liaison between these layers. A unified control plane would be useful to optimize the use of the network resources and automate the routing configurations. Software-Defined Networking (SDN) based architectures, such as OpenFlow, offer a chance to design such a control plane. One of the most important problems to deal with in this design is the path computation process. Classical path computation algorithms cannot resolve the problem as they do not take into account encapsulations and conversions of protocols. In this paper, we propose algorithms to solve this problem and study several cases: Path computation without bandwidth constraint, under bandwidth constraint and under other Quality of Service constraints. We study the complexity and the scalability of our algorithms and evaluate their performances on real topologies. The results show that they outperform the previous ones proposed in the literature.Comment: IEEE INFOCOM 2016, Apr 2016, San Francisco, United States. To be published in IEEE INFOCOM 2016, \<http://infocom2016.ieee-infocom.org/\&g

Network protection with multiple availability guarantees

We develop a novel network protection scheme that provides guarantees on both the fraction of time a flow has full connectivity, as well as a quantifiable minimum grade of service during downtimes. In particular, a flow can be below the full demand for at most a maximum fraction of time; then, it must still support at least a fraction q of the full demand. This is in contrast to current protection schemes that offer either availability-guarantees with no bandwidth guarantees during the downtime, or full protection schemes that offer 100% availability after a single link failure. We develop algorithms that provide multiple availability guarantees and show that significant capacity savings can be achieved as compared to full protection. If a connection is allowed to drop to 50% of its bandwidth for 1 out of every 20 failures, then a 24% reduction in spare capacity can be achieved over traditional full protection schemes. In addition, for the case of q = 0, corresponding to the standard availability constraint, an optimal pseudo-polynomial time algorithm is presented.National Science Foundation (U.S.) (NSF grants CNS-1116209)National Science Foundation (U.S.) (NSF grants CNS-0830961)United States. Defense Threat Reduction Agency (grant HDTRA-09-1-005)United States. Defense Threat Reduction Agency (grant HDTRA1-07-1-0004)United States. Air Force (Air Force contract # FA8721-05-C-0002

Optimizing busy time on parallel machines

We consider the following fundamental scheduling problem in which the input consists of n jobs to be scheduled on a set of identical machines of bounded capacity g (which is the maximal number of jobs that can be processed simultaneously by a single machine). Each job is associated with a start time and a completion time, it is supposed to be processed from the start time to the completion time (and in one of our extensions it has to be scheduled also in a continuous number of days, this corresponds to a two-dimensional version of the problem). We consider two versions of the problem. In the scheduling minimization version the goal is to minimize the total busy time of machines used to schedule all jobs. In the resource allocation maximization version the goal is to maximize the number of jobs that are scheduled for processing under a budget constraint given in terms of busy time. This is the first study of the maximization version of the problem. The minimization problem is known to be NP-Hard, thus the maximization problem is also NP-Hard. We consider various special cases, identify cases where an optimal solution can be computed in polynomial time, and mainly provide constant factor approximation algorithms for both minimization and maximization problems. Some of our results improve upon the best known results for this job scheduling problem. Our study has applications in power consumption, cloud computing and optimizing switching cost of optical networks

Traffic Grooming in Bidirectional WDM Ring Networks

We study the minimization of ADMs (Add-Drop Multiplexers) in optical WDM bidirectional rings considering symmetric shortest path routing and all-to-all unitary requests. We precisely formulate the problem in terms of graph decompositions, and state a general lower bound for all the values of the grooming factor $C$ and $N$, the size of the ring. We first study exhaustively the cases $C=1$, $C = 2$, and $C=3$, providing improved lower bounds, optimal constructions for several infinite families, as well as asymptotically optimal constructions and approximations. We then study the case $C>3$, focusing specifically on the case $C = k(k+1)/2$ for some $k \geq 1$. We give optimal decompositions for several congruence classes of $N$ using the existence of some combinatorial designs. We conclude with a comparison of the cost functions in unidirectional and bidirectional WDM rings

Traffic grooming in bidirectional WDM ring networks

International audienceWe study the minimization of ADMs (Add-Drop Multiplexers) in optical WDM bidirectional rings considering symmetric shortest path routing and all-to-all unitary requests. We precisely formulate the problem in terms of graph decompositions, and state a general lower bound for all the values of the grooming factor C and N, the size of the ring. We first study exhaustively the cases C = 1, C = 2, and C = 3, providing improved lower bounds, optimal constructions for several infinite families, as well as asymptotically optimal constructions and approximations. We then study the case C > 3, focusing specifically on the case C = k(k + 1)/2 for some k ≥ 1. We give optimal decompositions for several congruence classes of N using the existence of some combinatorial designs. We conclude with a comparison of the cost functions in unidirectional and bidirectional WDM rings