. A bidirected tree is the directed graph obtained from an undirected tree by replacing each undirected edge by two directed edges with opposite directions. Given a set of directed paths in a bidirected tree, the goal of the maximum edge-disjoint paths problem is to select a maximumcardinality subset of the paths such that the selected paths are edge-disjoint. This problem can be solved optimally in polynomial time for bidirected trees of constant degree, but is MAXSNP-hard for bidirected trees of arbitrary degree. For every fixed " ? 0, a polynomial-time (5=3+ ")-approximation algorithm is presented. Key words. approximation algorithms, edge-disjoint paths, bidirected trees AMS subject classifications. 68Q25, 68R10 1. Introduction. Research on disjoint paths problems in graphs has a long history [12]. In recent years, edge-disjoint paths problems have been brought into the focus of attention by advances in the field of communication networks. Many modern network architectures estab...

A very general technique for converting approximation algorithms for the vertex coloring problem in a class of graphs into approximation algorithms for the maximum weight independent set problem (MWIS) in the same class of graphs is presented. The technique consists of solving an LP-relaxation of the MWIS problem with certain clique inequalities, constructing an instance of the vertex coloring problem from the LP solution, applying the coloring algorithm to this instance, and selecting the best resulting color class as the MWIS solution. The approximation ratio obtained is the product of the approximation ratio with which the LP formulation can be solved (usually equal to one) and the approximation ratio of the coloring algorithm with respect to the size of the largest relevant clique. Applying this technique, the best known approximation algorithms are obtained for the maximum weight edge-disjoint paths problem in bidirected trees and in bidirected two-dimensional meshes ...

We consider the problem of assigning wavelengths to demands in an optical network of m links. We assume that the route of each demand is fixed and the number of wavelengths available on a fiber is some parameter . Our aim is to minimize the maximum ratio between the number of fibers deployed on a link e and the number of fibers required on the same link e when wavelength assignment is allowed to be fractional.

We study the problem of assigning a minimum number of colors to directed paths (dipaths) of a tree, so that any two dipaths that share a directed edge of the tree are not assigned the same color. The problem has applications to wavelength routing in WDM all-optical tree networks, an important engineering problem. Dipaths represent communication requests, while colors correspond to wavelengths that must be assigned to requests so that multiple users can communicate simultaneously through the same optical fiber. Recent work on wavelength routing in trees has studied a special class of algorithms which are called greedy. Although these algorithms are simple and implementable in a distributed setting, it has been proved that there are cases where a bandwidth utilization of 100% is not possible. Thus, in this work, we relax the constraints of the original engineering problem and use devices called wavelength converters that are able to convert the wavelength a...

We consider the problem of minimizing the amount of deployed fiber in optical networks in which each fiber carries a fixed number of wavelengths. We are given a network of general topology to carry a set of demands. For each demand we wish to choose a route and a wavelength. Since only distinct wavelengths can be carried on the same fiber, each link e requires max Fe () fibers where Fe () is the number of demands along e that are assigned wavelength . We wish to minimize the total amount of fiber deployed in order to carry all the demands. Most past work either assumed an unlimited number of wavelengths or else was restricted to specific topologies such as lines, rings and trees.

Abstract — We study the complexity of a spectrum of design problems for optical networks in order to carry a set of demands. Under wavelength division multiplexing (WDM) technology, demands sharing a common fiber are transported on distinct wavelengths. Multiple fibers may be deployed on a physical link. Our basic goal is to design networks of minimum cost, minimum congestion and maximum throughput. This translates to three variants in the design objectives: 1) MIN-SUMFIBER: minimizing the total amount of fibers deployed to carry all demands; 2) MIN-MAXFIBER: minimizing the maximum amount of fibers per link to carry all demands; and 3) MAX-THRUPUT: maximizing the carried demands using a given set of fibers. We also have two variants in the design constraints: 1) CHOOSEROUTE: Here we specify both a routing path and a wavelength for each demand; 2) FIXEDROUTE: Here we are given demand routes and we specify wavelengths only. The FIXEDROUTE variant allows us to study wavelength assignment in isolation. Combining these variants, we have six design problems. In [4], [3] we have shown that general instances of the problems MIN-SUMFIBER-CHOOSEROUTE and MIN-MAXFIBER-FIXEDROUTE have no constant-approximation algorithms. In this paper we prove that a similar statement holds for all four other problems. Our main result shows that MIN-SUMFIBER-FIXEDROUTE cannot be approximated within any constant factor unless NP-hard problems have efficient algorithms. This, together with the hardness of MIN-MAXFIBER-FIXEDROUTE in [3], shows that the problem of wavelength assignment is inherently hard by itself. We also study the complexity of problems that arise when multiple demands can be time-multiplexed onto a single wavelength (as in TWIN networks) and when wavelength converters can be placed along the path of a demand. I.

We study the problem of allocating optical bandwidth to sets of communication requests in all--optical networks that utilize Wavelength Division Multiplexing (WDM). WDM technology establishes communication between pairs of network nodes by establishing transmitter--receiver paths and assigning wavelengths to each path so that no two paths going through the same fiber link use the same wavelength. Optical bandwidth is the number of distinct wavelengths. Since state--of--the--art technology allows for a limited number of wavelengths, the engineering problem to be solved is to establish communication between pairs of nodes so that the total number of wavelengths used is minimized; this is known as the wavelength routing problem. In this paper, we survey recent advances in bandwidth allocation in tree--shaped WDM all-- optical networks: -- We present hardness results and lower bounds for the general problem and the special case of symmetric communication. -- We give the main ideas of deterministic greedy algorithms and study their limitations. -- We demonstrate how we can achieve optimal and nearly--optimal bandwidth utilization in networks with wavelength converters using simple algorithms. -- We also present recent results about the use of randomization for wavelength routing. 1

Abstract. We study computationally hard combinatorial problems arising from the important engineering question of how to maximize the number of connections that can be simultaneously served in a WDM optical network. In such networks, WDM technology can satisfy a set of connections by computing a route and assigning a wavelength to each connection so that no two connections routed through the same fiber are assigned the same wavelength. Each fiber supports a limited number of w wavelengths and in order to fully exploit the parallelism provided by the technology, one should select a set connections of maximum cardinality which can be satisfied using the available wavelengths. This is known as the maximum routing and path coloring problem (maxRPC). Our main contribution is a general analysis method for a class of iterative algorithms for a more general coloring problem. A lower bound on the benefit of such an algorithm in terms of the optimal benefit and the number of available wavelengths is given by a benefit-revealing linear program. We apply this method to maxRPC in both undirected and bidirected rings to obtain bounds on the approximability of several algorithms. Our results also apply to the problem maxPC where paths instead of connections are given as part of the input. We also study the profit version of maxPC in rings where each path has a profit and the objective is to satisfy a set of paths of maximum total profit.

A wavelength division multiplexed (WDM) network employs multiple wavelengths in order to carry many channels in an optical fiber. Switching is, ideally, done by a WDM cross-connect that allows each incoming input channel to be routed to any (unused) output channel. To do this the cross-connect requires expensive components called wavelength interchangers that permute the wavelengths on a fiber in any desired manner. The cross-connect is said to be wide-sense nonblocking if there is an on-line algorithm that assures that it can always meet demands. (This is weaker than strictly nonblocking where the demands are never blocked even when previous demands have been routed arbitrarily). Our goal here is to minimize the number of wavelength interchangers in the design of a wide-sense nonblocking cross-connect with k input fibers and k output fibers. It is easily seen that 2k\Gamma 1 wavelength interchangers suffice, even with greedy routing; we show that with O(k2) wavelengths they are necessary as well, regardless of the routing algorithm. This improves previous exponential bounds. On the positive side, we show that in the case where there are only 2 or 3 wavelengths there is a significant reduction in the number of wavelength interchangers required. However, we also show that for any " ? 0 and k? 1=2", if there are at least 1="2 wavelengths then 2(1\Gamma ")k wavelength interchangers are necessary. This WDM cross-connect problem is shown to be equivalent to a dynamic edge coloring problem for bipartite multigraphs and the results are stated and derived in terms of this edge coloring problem.

ity. We study wavelength assignment in an optical network where each fiber has a fixed capacity of µ wavelengths. Given demand routes, we aim to minimize the maximum ratio between the number of fibers deployed on a link e and the number of fibers required on the same link e when wavelength assignment is allowed to be fractional. Our main results are negative ones. We show that there is no constant-factor approximation unless NP⊆ZPP. In addition, unless NP ⊆ ZPTIME(n polylog n) we show that there is no log γ µ approximation for any γ ∈ (0, 1) and no log γ m approximation for any γ ∈ (0, 0.5) where m is the number of links in the network. Our analysis is based on the hardness of approximating the chromatic numbers. On the positive side, we present algorithms with approximation ratios O(log m+log µ), O(log Dmax +log µ) and O(Dmax) respectively, where Dmax is the length of the longest path.