Iterative Linear Programming Formulation for Routing and ...

.CADLogical LinksFig. 2: Logical topology embedded in Fig.1.i) Determine a good virtual topology, that is, theorigin and destination node of every lightpath.ii) Route the lightpaths over the physicaltopology.iii) Assign wavelengths to the lightpaths.iv) Route the externally offered traffic on thevirtual topology.In this study, we center in subproblems ii) andiii), which are known in joint as the Routing andWavelength Assignment (RWA) problem. It can bedefined as follows [2]: Given a network topology anda set of lightpath requests, determine a route andwavelength(s) for the requests. Some parameters to beoptimized when solving the RWA problem are thenumber of wavelengths needed, the wavelength reuse,the physical distance traversed by the traffic, and thecost.We propose two separate formulations to solvethe RWA problem when there is no wavelengthconversion. The former routes the lightpaths in a waythat the average distance traversed per traffic unit isminimized. Besides, it limits the number of lightpathsper fiber link (which is a heuristic for limiting thenumber of wavelengths needed). The latter assignswavelengths to the lightpaths once they have beenrouted. In this case, the objective is to maximize thewavelength reuse. We also propose an iterativealgorithm to solve the RWA problem by combiningboth formulations. Finally, a number of results areshowed.The lightpaths are routed using a shortest-pathalgorithm. Then, wavelengths are assigned to thoselightpaths with a large traffic load first.The routing problem is also solved by means of ashortest-path algorithm in [8], but wavelengths areassigned sequentially beginning with the longestlightpath.In [9], subproblem ii) is formulated as an integerlinear programming (ILP) problem. Severaltechniques, as randomized rounding, are employed toreduce the problem complexity. To solve subproblemiii), graph-coloring algorithms are used.A mixed integer linear programming (MILP)formulation for subproblems i) and ii), as well as anumber of heuristics for solving subproblem i) injoint with the RWA problem are proposed in [5].In [10], an ILP formulation and two heuristics(based on Dijkstra’s algorithm) are proposed forsolving RWA. One of them routes lightpathssequentially. The other one routes them in parallel.Both take into account the available wavelengths andtheir capacity at every fiber link. A cost model isstated to compute the distance used by the shortestpathalgorithm. Finally, an improvement step is run.A genetic-algorithm/heuristic hybrid approach isproposed in [11] for solving RWA with the objectiveof minimizing the cost of the network.3 Routing and Wavelength AssignmentAs stated before, the RWA problem consists inrouting the lightpaths through the physical topologyand assigning wavelengths to them. We suppose thereis no wavelength conversion, so each lightpath isassigned a single wavelength. In the next paragraphs,we propose new separate ILP formulations for solvingsubproblems ii) and iii). By separating RWA intosubproblems the total computational cost is reduced.2 Previous WorkIn the last years, several papers about virtualtopology design have been published [4−17]. Next,we will overview some of them concerning therouting and/or the wavelength assignment problem.The work in [4] proposes to allocate thewavelengths to the longest lightpaths first, as aheuristic for minimizing the number of wavelengthsneeded when solving subproblem iii).In [7], subproblems i), ii) and iii) are considered.3.1 The Lightpath Routing ProblemIn this problem, a route over the physical topology iscomputed for every lightpath. As proposed in [6] and[11], we use flow conservation equations at each nodeto properly route the lightpaths. Note that the numberof wavelengths that are needed for establishing thelightpaths, once they have been routed, depends onthe maximum number of lightpaths that share a singlefiber link (we call it ρ max ). In fact, ρ max will be a lowerbound on the number of wavelengths. In Fig. 1, the