The construction of disjoint paths in a network is a basic issue in combinatorial optimization: given a network, and specified pairs of nodes in it, we are interested in finding disjoint paths between as many of these pairs as possible. This leads to a variety of classical NP-complete problems for which very little is known from the point of view of approximation algorithms. It has recently been brought into focus in work on problems such as VLSI layout and routing in high-speed networks; in these settings, the current lack of understanding of the disjoint paths problem is often an obstacle to the design of practical heuristics.

We deal with randomized competitive algorithms for non-preemptive call control on tree-like switching networks. We give an optimal O(log n) competitive algorithm for non-preemptive call scheduling on trees. We then extend the problem to include variable call rates, call durations, and arbitrary call benefits, and obtain a polylog competitive algorithm. We also show that many similar algorithms for different problems that can deal with constant values of parameters such as rates and benefits can be transformed into randomized algorithms that can deal with varying values of the parameters.

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.

In this paper we study the problem of on-line allocation of routes to virtual circuits (both point-topoint and multicast) where the goal is to minimize the required bandwidth. We concentrate on the case of permanent virtual circuits (i.e., once a circuit is established, it exists forever), and describe an algorithm that achieves an O(log n) competitive ratio with respect to maximum congestion, where n is the number of nodes in the network. Informally, our results show that instead of knowing all of the future requests, it is sufficient to increase the bandwidth of the communication links by an O(log n) factor. We also show that this result is tight, i.e. for any on-line algorithm there exists a scenario in which O(log n) increase in bandwidth is necessary. We view virtual circuit routing as a generalization of an on-line load balancing problem, defined as follows: jobs arrive on line and each job must be assigned to one of the machines immediately upon arrival. Assigning a job to a machine increases this machine’s load by an amount that depends both on the job and on the machine. The goal is to minimize the maximum load. For the related machines case, we describe the first algorithm that achieves constant competitive ratio. For the unrelated case (with n machines), we describe a new method that yields O(log n)-competitive

One of the central tasks of networking is packet-routing when edge bandwidth is limited. Tremendous progress has been achieved by separating the issue of routing into two conceptual sub-problems: path selection and congestion resolution along the selected paths. However, this conceptual separation has a serious drawback: each packet's path is fixed at the source and cannot be modified adaptively en-route. The problem is especially severe when packet injections are modeled by an adversary, whose goal is to cause "traffic-jams".

We present lower bounds on the competitive ratio of randomized algorithms for a wide class of on-line graph optimization problems and we apply such results to on-line virtual circuit and optical routing problems. Lund and Yannakakis [LY93a] give inapproximability results for the problem of finding the largest vertex induced subgraph satisfying any non-trivial, hereditary, property . E.g., independent set, planar, acyclic, bipartite, etc. We consider the on-line version of this family of problems, where some graph G is fixed and some subgraph H is presented on-line, vertex by vertex. The on-line algorithm must choose a subset of the vertices of H , choosing or rejecting a vertex when it is presented, whose vertex induced subgraph satisfies property . Furthermore, we study the on-line version of graph coloring whose off-line version has also been shown to be inapproximable [LY93b], on-line max edge-disjoint paths and on-line path coloring problems. Irrespective of the time complexity, w...

In this paper we consider the unsplittable ow problem (UFP): given a directed or undirected network G = (V, E) with edge capacities and a set of terminal pairs (or requests) with associated demands, find a subset of the pairs of maximum total demand for which a single flow path can be chosen for each pair so that for every edge, the sum of the demands of the paths crossing the edge does not exceed its capacity.

The max-flow min-cut theorem of Ford and Fulkerson is based on an even more foundational result, namely Menger's theorem on graph connectivity. Menger's theorem provides a good characterization for the following single-source disjoint paths problem: given a graph G, with a source vertex s and terminals t 1 , ..., t k , decide whether there exist edge-disjoint s-t i paths, for i = 1, ..., k. We consider a natural, NP-hard generalization of this problem, which we call the single-source unsplittable flow problem. We are given a source and terminals as before; but now each terminal t i has a demand ae i 1, and each edge e of G has a capacity c e 1. The problem is to decide whether one can choose a single s-t i path, for each i, so that the resulting set of paths respects the capacity constraints --- the total amount of demand routed across any edge e must be bounded by the capacity c e . The main results of this paper are constant-factor approximation algorithms for three n...

We study the on-line call admission problem in optical networks. We present a general technique that allows us to reduce the problem of call admission and wavelength selection to the call admission problem. We then give randomized algorithms with logarithmic competitive ratios for specific topologies in switchless and reconfigurable optical networks. We conclude by considering full duplex communications.

Emerging high speed networks will carry traffic for services such as video-on-demand and video teleconferencing -- that require resource reservation along the path on which the traffic is sent. High bandwidth-delay product of these networks prevents circuit rerouting, i.e. once a circuit is routed on a certain path, the bandwidth taken by this circuit remains unavailable for the duration (holding time) of this circuit. As a result, such networks will need effective routing and admission control strategies. Recently developed online routing and admission control strategies have logarithmic competitive ratios with respect to the admission ratio (the fraction of admitted circuits). Such guarantees on performance are rather weak in the most interesting case where the rejection ratio of the optimum algorithm is very small or even 0. Unfortunately, these guarantees can not be improved in the context of the considered models, making it impossible to use these models to identify algorithms th...