. Motivated by the problem of efficient routing in all-optical networks, we study a constrained version of the bipartite edge coloring problem. We show that if the edges adjacent to a pair of opposite vertices of an L-regular bipartite graph are already colored with ffL different colors, then the rest of the edges can be colored using at most (1+ff=2)L colors. We also show that this bound is tight by constructing instances in which (1 + ff=2)L colors are indeed necessary. We also obtain tight bounds on the number of colors that each pair of opposite vertices can see. Using the above results, we obtain a polynomial time greedy algorithm that assigns proper wavelengths to a set of requests of maximum load L per directed fiber link on a directed fiber tree using at most 5=3L wavelengths. This improves previous results of [9, 7, 6, 10]. We also obtain that no greedy algorithm can in general use less than 5=3L wavelengths for a set of requests of load L in a directed fiber tree, and thus t...

Abstract not found

The paper deals with on-line routing in WDM (wavelength division multiplexing) optical networks. A sequence of requests arrives over time, each is a pair of nodes to be connected by a path. The problem is to assign a wavelength and a path to each pair, so that no two paths sharing a link are assigned the same wavelength. The goal is to minimize the number of wavelengths used to establish all connections. Raghavan and Upfal [RU94] considered the off-line version of the problem, which was further studied in [AR95, KP96, MKR95, Ra96]. For a line topology, the problem is the well-studied interval graph coloring problem. On-line algorithms for this problem have been analyzed in [KT81, Ki88]. We consider trees, trees of rings, and meshes topologies, previously studied in the off-line case. We give on-line algorithms with competitive ratio O(log n) for all these topologies. We give a matching \Omega\Gammaing n) lower bound for meshes. We also prove that any algorithm for trees canno...

Abstract In this paper we study wavelength assignment on an optical linesystem without wavelength conversion. Consider a set of undirected demands along the line. Each demand is carried on a wavelength and any two overlapping demands require distinct wavelengths. Suppose _ wavelengths are available in the system. We define `(e), the load on link e, to be the smallest integer such that `(e) _ is at least the number of demands passing through e. Hence, `(e) is the minimum number of fibers required on e in order to support all demands. We present a polynomial-time wavelength assignment algorithm that guarantees each wavelength appears at most `(e) times on each link e. (This generalizes the well-known fact that interval graphs are perfect.) In the presence of MOADMs (mesh optical add/drop multiplexers), devices that multiplex distinct wavelengths from different fibers into a new fiber, we only need to deploy `(e) fibers per link. On the other hand, if each demand has to stay on a single fiber, as is the case without MOADMs, we show that some links may require more than `(e) fibers. In fact, we show that it is NPcomplete to decide if a set of demands can be carried on a given set of fibers, or if there exists a set of fibers with a given total length that can carry all the demands.

Modern high-performance communication networks pose a number of challenging problems concerning the efficient allocation of resources to connection requests. In all-optical networks with wavelength-division multiplexing, connection requests must be assigned paths and colors (wavelengths) such that intersecting paths receive different colors, and the goal is to minimize the number of colors used. This path coloring problem is proved NP-hard for undirected and bidirected ring networks. Path coloring in undirected tree networks is shown to be equivalent to edge coloring of multigraphs, which implies a polynomial-time optimal algorithm for trees of constant degree as well as NP-hardness and an approximation algorithm with absolute approximation ratio 4:3 and asymptotic approximation ratio 1:1 for trees of arbitrary degree. For bidirected trees, path coloring is shown to be NP-hard even in the binary case. A polynomial-time optimal algorithm is given for path coloring in undirected or bidir...

. 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...

. In optical networks with wavelength division multiplexing (WDM), multiple connections can share a link if they are transmitted on different wavelengths. We study the problem of satisfying a maximum number of connection requests in a directed tree network if only a limited number W of wavelengths are available. In optical networks without wavelength converters this is the maximum path coloring (MaxPC) problem, in networks with full wavelength conversion this is the maximum path packing (MaxPP) problem. MaxPC and MaxPP are shown to be polynomial-time solvable to optimality if the tree has height one or if both W and the degree of the tree are bounded by a constant. If either W or the degree of the tree is not bounded by a constant, MaxPC and MaxPP are proved NP-hard. Polynomial-time approximation algorithms with performance ratio 5=3 + " for arbitrarily small " are presented for the case W = 1, in which MaxPC and MaxPP are equivalent. For arbitrary W , a 2-approximation for MaxPP in arbitrary trees, a 1:58-approximation for MaxPC in trees of bounded degree, and a 2:22-approximation for MaxPC in arbitrary trees are obtained. 1

. Motivated by the problem of WDM routing in all--optical networks, we study the following NP--hard problem. We are given a directed binary tree T and a set R of directed paths on T . We wish to assign colors to paths in R, in such a way that no two paths that share a directed arc of T are assigned the same color and that the total number of colors used is minimized. Our results are expressed in terms of the depth of the tree and the maximum load l of R, i.e., the maximum number of paths that go through a directed arc of T . So far, only deterministic greedy algorithms have been presented for the problem. The best known algorithm colors any set R of maximum load l using at most 5l=3 colors. Alternatively, we say that this algorithm has performance ratio 5=3. It is also known that no deterministic greedy algorithm can achieve a performance ratio better than 5=3. In this paper we define the class of greedy algorithms that use randomization. We study their limitations and pr...

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.

) Luisa Gargano Dipartimento di Informatica ed Applicazioni Universit`a di Salerno 84081 Baronissi (SA), Italy. Abstract Let T be a symmetric directed tree, i.e., a tree with each edge viewed as two opposite directed links. We consider the problem of routing arbitrary sets of connection requests in T . In all-optical communication tree networks with WDM (wavelength-division multiplexing) this is equivalent to color (assign wavelengths to) a given set of directed paths so that no two directed paths of the same color use the same link of T . Let W be the number of available wavelengths. The load, that is, the maximum number of directed paths passing through a link of T cannot exceed W . If there is no wavelength conversion available then each request (directed path) is restricted to a single wavelength and it is known that the minimum number of colors needed to color any set of directed paths in a tree is lower bounded away from the load L of the paths on the tree; moreover, no algori...