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".

In heterogeneous networks, sending messages may incur different delays on different links, and each node may have a different switching time between messages. The well studied Telephone model is obtained when all link delays and switching times are equal to one unit. We investigate the problem of finding the minimum time required to multicast a message from one source to a subset of the nodes of size k. The problem is NP-hard even in the basic Telephone model. We present a polynomial time algorithm that approximates the minimum multicast time within a factor of O(log k). Our algorithm improves on the best known approximation factor for the Telephone model by a factor of O log n log log k . No approximation algorithms were known for the general model considered in this paper.

We propose asymptotically optimal algorithms for the job shop scheduling and packet routing problems. We propose a fluid relaxation for the job shop scheduling problem in which we replace discrete jobs with the flow of a continuous fluid. We compute an optimal solution of the fluid relaxation in closed form, obtain a lower bound Cmax to the job shop scheduling problem, and construct a feasible schedule from the fluid relaxation with objective value at most C � OŽ C. max ' max, where the constant in the OŽ. � notation is independent of the number of jobs, but it depends on the processing time of the jobs, thus producing an asymptotically optimal schedule as the total number of jobs tends to infinity. If the initially present jobs increase proportionally, then our algorithm produces a schedule with value at most C � OŽ. max 1. For the packet routing problem with fixed paths the previous algo-rithm applies directly. For the general packet routing problem we propose a linear programming relaxation that provides a lower bound Cmax and an asymptotically optimal algorithm that uses the optimal solution of the relaxation with objective value at most C � OŽ C. max ' max. Unlike asymptotically optimal algorithms that rely on probabilistic assumptions, our proposed algorithms make no probabilistic assumptions and they are asymptotically optimal for all instances with a large number of jobs Ž packets.. In computational experiments our algorithms produce schedules which are within 1 % of optimality even for moderately sized problems.

We consider the problem of designing a minimum cost access network to carry traffic from a set of endnodes to a core network. A set of trunks of K differing types are available for leasing or buying. Some trunk-types have a high initial overhead cost but a low cost per unit bandwidth. Others have a low overhead cost but a high cost per unit bandwidth. When the central core is given, we show how to construct an access network whose cost is within O(K 2 ) of optimal, under weak assumptions on the cost structure. In contrast with previous bounds, this bound is independent of the network and the traffic. Typically, the value of K is small. Our approach uses a linear programming relaxation and is motivated by a rounding technique of Shmoys, Tardos and Aardal [15]. Our techniques extend to a more complex situation in which the core is not given a priori. In this case we aim to minimize the switch cost of the core in addition to the trunk cost of the access network. We provide the same pe...

We investigate stability of packet routing policies in adversarial queueing networks. We provide a simple classification of networks which are stable under any greedy scheduling policy - network is stable if and only if the underlying undirected connected graph contains at most two edges. We also propose a simple and distributed policy which is stable in an arbitrary adversarial queueing network even for the critical value of the arrival rate r = 1. Finally, a simple and checkable network flow type load condition is formulated for adaptive adversarial queueing networks and a policy is proposed which achieves stability under this new load condition. This load condition is a relaxation of the integral network flow type condition considered previously in the literature.

William Aiello Rafail Ostrovsky Eyal Kushilevitz Adi Ros'en Abstract The combination of the buffer size of routers deployed in the Internet and the Internet traffic itself leads routinely to routers dropping packets. Motivated by this, we initiate the rigorous study of dynamic storeand -forward routing on arbitrary networks in a model in which dropped packets must explicitly be taken into account. To avoid the uncertainties of traffic modeling, we consider arbitrary traffic on the network. We analyze and compare the effectiveness of several greedy, on-line, local-control protocols using a competitive analysis of the throughput. One goal of our approach is for the competitive results to continue to hold as a network grows without requiring the memory in the nodes to increase with the size of the network. Thus, in our model, we have link buffers of fixed size, B, which is independent of the size of the network, and B becomes a parameter of the model.

In this paper we study approximation algorithms for solving a general covering integer program. An n-vector x of nonnegative integers is sought, which minimizes c T x; subject to Ax b; x d: The entries of A; b; c are nonnegative. Let m be the number of rows of A: Covering problems have been heavily studied in combinatorial optimization. We focus on the effect of the multiplicity constraints, x d; on approximability. Two longstanding open questions remain for this general formulation with upper bounds on the variables. (i) The integrality gap of the standard LP relaxation is arbitrarily large. Existing approximation algorithms that achieve the well-known O(log m)-approximation with respect to the LP value do so at the expense of violating the upper bounds on the variables by the same O(log m) multiplicative factor. What is the smallest possible violation of the upper bounds that still achieves cost within O(log m) of the standard LP optimum ? (ii) The best known approximation ratio for the problem has been O(log(max j P i A ij )) since 1982. This bound can be as bad as polynomial in the input size. Is an O(log m)-approximation, like the one known for the special case of Set Cover, possible? We settle these two open questions. To answer the first question we give an algorithm based on the relatively simple new idea of randomly rounding variables to smaller-thaninteger units. To settle the second question we give a reduction from approximating the problem while respecting multiplicity constraints to approximating the problem with a bounded violation of the multiplicity constraints. 1 Research partially supported by NSERC Grant 227809-00 and a CFI New Opportunities Award 1.

We study routing and scheduling in packet-switched networks. We assume an adversary that controls the injection time, source, and destination for each packet injected. A set of paths for these packets is admissible if no link in the network is overloaded. We present the first on-line routing algorithm that finds a set of admissible paths whenever this is feasible. Our algorithm calculates a path for each packet as soon as it is injected at its source using a simple shortest path computation. The length of a link reflects its current congestion. We also show how our algorithm can be implemented under today's Internet routing paradigms.

Given matrices A and B and vectors a, b, c and d, all with non-negative entries, we consider the problem of computing min{cT x: x ∈ Z n +,Ax�a, Bx �b, x �d}. We give a bicriteria-approximation algorithm that, given � ∈ (0, 1], finds a solution of cost O(ln(m)/ � 2) times optimal, meeting the covering constraints (Ax �a) and multiplicity constraints (x �d), and satisfying Bx�(1 + �)b + �, where � is the vector of row sums �i = � j Bij. Here m denotes the number of rows of A. This gives an O(ln m)-approximation algorithm for CIP—minimum-cost covering integer programs with multiplicity constraints, i.e., the special case when there are no packing constraints Bx�b. The previous best approximation ratio has been O(ln(maxj iAij)) since 1982. CIP contains the set cover problem as a special case, so O(ln m)-approximation is the best possible unless P = NP.

In 1988, Leighton, Maggs, and Rao showed that for any network and any set of packets whose paths through the network are fixed and edge-simple, there exists a schedule for routing the packets to their desti-nations in O(c + d) steps using constant-size queues, where c is the congestion of the paths in the network, and d is the length of the longest path. The proof, however, used the Lovdsz Local Lemma and was not constructive. In this paper, we show how to find such a schedule in O(NE + E 1og’E) time, for any fixed 6> 0, where N is the total number of packets, and E is the number of edges in the network. We also show how to parallelize the algorithm so that it runs in NC. The method that we use to construct eficient packet routing schedules is based on the algorithmic form of the Lovdsz Local Lemma discovered by Beck. 1