We study the approximability of edge-disjoint paths and related problems. In the edge-disjoint paths problem (EDP), we are given a network G with source-sink pairs (si, ti), 1 ≤ i ≤ k, and the goal is to find a largest subset of source-sink pairs that can be simultaneously connected in an edge-disjoint manner. We show that in directed networks, for any ɛ> 0, EDP is NP-hard to approximate within m 1/2−ɛ. We also design simple approximation algorithms that achieve essentially matching approximation guarantees for some generalizations of EDP. Another related class of routing problems that we study concerns EDP with the additional constraint that the routing paths be of bounded length. We show that, for any ɛ> 0, bounded length EDP is hard to approximate within m 1/2−ɛ even in undirected networks, and give an O ( √ m)-approximation algorithm for it. For directed networks, we show that even the single source-sink pair case (i.e. find the maximum number of paths of bounded length between a given sourcesink pair) is hard to approximate within m 1/2−ɛ, for any ɛ> 0.

We survey the theoretical results obtained for wavelength routing in all–optical networks, present some new results and propose several open problems. In all–optical networks the vast bandwidth available is utilized through wavelength division multiplexing: a single physical optical link can carry several logical signals, provided that they are transmitted on different wavelengths. The information, once transmitted as light, reaches its destination without being converted to electronic form in between, thus reaching high data transmission rates. We consider both networks with arbitrary topologies and particular networks of practical interest.

This paper studies the problems of broadcasting and gossiping in optical networks. In such networks the vast bandwidth available is utilized through wavelength division multiplexing: a single physical optical link can carry several logical signals, provided that they are transmitted on different wavelengths. In this paper we consider both single--hop and multihop optical networks. In single--hop networks the information, once transmitted as light, reaches its destination without being converted to electronic form in between, thus reaching high speed communication. In multi hop networks a packet may have to be routed through a few intermediate nodes before reaching its final destination. In both models, we give efficient broadcasting and gossiping algorithms, in terms of time and number of wavelengths. We consider both networks with arbitrary topologies and particular networks of practical interest. Several of our algorithms exhibit optimal performances. 1 Introduction Motivations. Op...

Abstract In the Edge-Disjoint Paths problem with Congestion(EDPwC), we are given a graph with n nodes, a set of ter-minal pairs and an integer c. The objective is to route asmany terminal pairs as possible, subject to the constraint that at most c demands can be routed through any edge inthe graph. When c = 1, the problem is simply referred to asthe Edge-Disjoint Paths (EDP) problem. In this paper, we study the hardness of EDPwC in undirected graphs.We obtain an improved hardness result for EDP, and also show the first polylogarithmic integrality gaps andhardness of approximation results for EDPwC. Specifically, we prove that EDP is (log 12- " n)-hard to approximate for any constant "> 0, unless N P ` ZP T IME(npolylog n). We also show that for any congestion c = o(log log n / log log log n), there is no (log

In the minimum path coloring problem, we are given a list of pairs of vertices of a graph. We are asked to connect each pair by a colored path. Paths of the same color must be edge disjoint. Our objective is to minimize the number of colors used. This problem was raised by Aggarwal et al [1] and Raghavan and Upfal [22] as a model for routing in all-optical networks. It is also related to questions in circuit routing. In this paper, we improve the O(ln N ) approximation result of Kleinberg and Tardos [14] for path coloring on the N \Theta N mesh. We give an O(1) approximation algorithm to the number of colors needed, and a poly(ln ln N ) approximation algorithm to the choice of paths and colors. To the best of our knowledge, these are the first sub-logarithmic bounds for any network other than trees, rings, or trees of rings. Our results are based on developing new techniques for randomized rounding. These techniques iteratively improve a fractional solution until it approaches integral...

This paper studies the problem of All-to-All Communication for optical networks. In such networks the vast bandwidth available is utilized through wavelength division multiplexing (WDM): a single physical optical link can carry several logical signals, provided that they are transmitted on different wavelengths. In this paper we consider all-optical (or singlehop) networks, where the information, once transmitted as light, reaches its destination without being converted to electronic form in between, thus reaching high data transmission rates. In this model, we give optimal all-to-all protocols, using minimum numbers of wavelengths, for particular networks of practical interest, namely the d-dimensional square tori with even side, the corresponding meshes and the Cartesian sums of complete graphs.

We consider the maximum disjoint paths problem and its generalization, the call control problem, in the on-line setting. In the maximum disjoint paths problem, we are given a sequence of connection requests for some communication network. Each request consists of a pair of nodes, that wish to communicate over a path in the network. The request has to be immediately connected or rejected, and the goal is to maximize the number of connected pairs, such that no two paths share an edge. In the call control problem, each request has an additional bandwidth speci cation, and the goal is to maximize the total bandwidth of the connected pairs (throughput), while satisfying the bandwidth constraints (assuming each edge has unit capacity). These classical problems are central in routing and admission control in high speed networks and in optical networks.

Bandwidth is a very valuable resource in wavelength division multiplexed optical networks. The problem of finding an optimal assignment of wavelengths to requests is of fundamental importance in bandwidth utilization. We present a polynomialtime algorithm for this problem on fixed constant-size topologies. We combine this algorithm with ideas from Raghavan and Upfal [15] to obtain an optimal assignment of wavelengths on constant degree undirected trees. Mihail, Kaklamanis, and Rao [14] posed the following open question: what is the complexity of this problem on directed trees? We show that it is NP-complete both on binary and constant depth directed trees. Keywords: Algorithms, Combinatorial Problems, Computational Complexity, Interconnection Networks. 1 Introduction Motivation. Developments in fiber-optic networking technology using Wavelength Division Multiplexing (WDM) have finally reached the point where it 1 Supported by ONR Young Investigator Award N00014-93-1-0590. This work ...

In this paper we study the classical problem of finding disjoint paths in graphs. This problem has been studied by a number of authors both for specific graphs and general classes of graphs. Whereas for specific graphs many (almost) matching upper and lower bounds are known for the competitiveness of on-line algorithms, not much is known about how well on-line algorithms can perform in the general setting. In several papers the expansion has been used to measure the performance of off-line and on-line algorithms in this field. We study a class of simple deterministic on-line algorithms, called bounded greedy algorithms, and show that they achieve a competitive ratio that is asymptotically equal to the best possible competitive ratio that can be achieved by any deterministic on-line algorithm. For this we use a parameter called routing number that allows more precise results than the expansion. Interestingly, our upper bound on the competitive ratio is even better than the best approximation ratio known for off-line algorithms. Furthermore, we introduce a refined variant of the routing number and show that this variant allows to construct online algorithms with a competitive ratio that can be significantly below the best possible upper bound for deterministic on-line algorithms if only the routing number or expansion of a network is known. We also show that our on-line algorithms can be transformed into efficient algorithms for the related unsplittable flow problem.

. Let T be a symmetric directed tree, i.e., an undirected tree with each edge viewed as two opposite arcs. We prove that the minimum number of colours needed to colour the set of all directed paths in T , so that no two paths of the same colour use the same arc of T , is equal to the maximum number of paths passing through an arc of T . This result is applied to solve the all-to-all communication problem in wavelength-- division--multiplexing (WDM) routing in all--optical networks, that is, we give an efficient algorithm to optimally assign wavelengths to the all the paths of a tree network. It is known that the problem of colouring a general subset of all possible paths in a symmetric directed tree is an NPhard problem. We study conditions for a given set S of paths be coloured efficiently with the minimum possible number of colours/wavelengths. 1 Introduction Let T be a tree and x; y two vertices of T . The dipath P (x; y) in T is the undirected path joining x to y, in which each ed...