We describe fully polynomial time approximation schemes for various multicommodity flow problems in graphs with m edges and n vertices. We present the first approximation scheme for maximum multicommodity flow that is independent of the number of commodities k, and our algorithm improves upon the runtime of previous algorithms by this factor of k, performing in O (ffl \Gamma2 m 2 ) time. For maximum concurrent flow, and minimum cost concurrent flow, we present algorithms that are faster than the current known algorithms when the graph is sparse or the number of commodities k is large, i.e. k ? m=n. Our algorithms build on the framework proposed by Garg and Konemann [4]. They are simple, deterministic, and for the versions without costs, they are strongly polynomial. Our maximum multicommodity flow algorithm extends to an approximation scheme for the maximum weighted multicommodity flow, which is faster than those implied by previous algorithms by a factor of k= log W where W is ...

After several decades of sustained research and testing, linear programming has evolved into a remarkably reliable, accurate and useful tool for handling industrial optimization problems. Yet, large problems arising from several concrete applications routinely defeat the very best linear programming codes, running on the fastest computing hardware. Moreover, this is a trend that may well continue and intensify, as problem sizes escalate and the need for fast algorithms becomes more stringent. Traditionally, the focus in optimization algorithms, and in particular, in algorithms for linear programming, has been to solve problems "to optimality." In concrete implementations, this has always meant the solution ofproblems to some finite accuracy (for example, eight digits). An alternative approach would be to explicitly, and rigorously, trade o# accuracy for speed. One motivating factor is that in many practical applications, quickly obtaining a partially accurate solution is much preferable to obtaining a very accurate solution very slowly. A secondary (and independent) consideration is that the input data in many practical applications has limited accuracy to begin with. During the last ten years, a new body ofresearch has emerged, which seeks to develop provably good approximation algorithms for classes of linear programming problems. This work both has roots in fundamental areas of mathematical programming and is also framed in the context ofthe modern theory ofalgorithms. The result ofthis work has been a family ofalgorithms with solid theoretical foundations and with growing experimental success. In this manuscript we will study these algorithms, starting with some ofthe very earliest examples, and through the latest theoretical and computational developments.

We present a survey of nondifferentiable optimization problems and methods with special focus on the analytic center cutting plane method. We propose a self-contained convergence analysis, that uses the formalism of the theory of self-concordant functions, but for the main results, we give direct proofs based on the properties of the logarithmic function. We also provide an in depth analysis of two extensions that are very relevant to practical problems: the case of multiple cuts and the case of deep cuts. We further examine extensions to problems including feasible sets partially described by an explicit barrier function, and to the case of nonlinear cuts. Finally, we review several implementation issues and discuss some applications.

In this paper, we propose and analyze three routing algorithms for anycast packets: i) source-destination based routing with weighted random selection (SD/WRS), ii) destination based routing with weighted random selection (D/WRS), and iii) the shortest shortest path first (SSPF) algorithms. The SSPF algorithm is a simple extension to the traditional SPF algorithm for routing unicast packets. The SD/WRS and D/WRS algorithms explicitly take into account characteristics of anycast message traffic and its recipient group. As a result, our simulation study shows that both the SD/WRS and D/WRS algorithms perform much better than SSPF in terms of average end-to-end packet delay. In particular, SD/WRS performs very close to a dynamic optimal algorithm in most cases. Our algorithms are simple, efficient, and compatible with the most of existing routing technologies. We also formally prove the loop free and correctness properties for our algorithms.

AbstractÐAn anycast packet is one that should be delivered to one member in a group of designated recipients. Using anycast services may considerably simplify some applications. Little work has been done on routing anycast packets. In this paper, we propose and analyze a routing protocol for anycast message. It is composed of two subprotocols: the routing table establishment subprotocol and the packet forwarding subprotocol. In the routing table establishment subprotocol, we propose four methods (SSP, MIN-D, SBT, and CBT) for enforcing an order among routers for the purpose of loop prevention. These methods differ from each other on information used to maintain orders, the impact on QoS, and the compatibility to the existing routing protocols. In the packet forwarding subprotocol, we propose a Weighted-Random Selection (WRS) approach for multiple path selection in order to balance network traffic. In particular, the fixed and adaptive methods are proposed to determine the weights. Both of them explicitly take into account the characteristics of distribution of anycast recipient group while the adaptive method uses the dynamic information of the anycast traffic as well. Correctness property of the protocol is formally proven. Extensive simulation is performed to evaluate our newly designed protocol. Performance data shows that the loop-prevention methods and the WRS approaches have great impact on the performance in terms of average end-to-end packet delay. In particular, the protocol using the SBT or CBT loop-prevention methods and the adaptive WRS approach performs very close to a dynamic optimal routing protocol in most cases. Index TermsÐAnycast message, multiple path routing, shortest path first, weight assignment. 1

In this paper, we consider solution approaches to a multi-hour combined capacity design and routing problem which arises in the design of dynamically reconfigurable broadband communication networks that uses the virtual path concept. We present a comparative evaluation of four approaches, namely: a genetic algorithm; a Lagrangean relaxation based subgradient optimization method; a generalized proximal point algorithm with subgradient optimization; and finally, a hybrid approach where the subgradient based method is combined with a genetic algorithm. Our computational experience on a set of test problems of varying network sizes shows that the hybrid approach often is the desirable choice in obtaining the minimum cost network while the genetic algorithm based approach has the most difficulty in solving large scale problems. Keywords: Broadband network design, multi-hour network routing and capacity design, combinatorial optimization, genetic algorithm, duality and subgradient optimizati...

The increasing application space of interconnection networks now encompasses several applications, such as packet routing and I/O interconnect, where the throughput of a routing algorithm, not just its locality, becomes an important performance metric. We show that the problem of designing oblivious routing algorithms that have high worst-case or average-case throughput can be cast as a linear program. Globally optimal solutions to these optimization problems can be efficiently found, yielding provably good oblivious routing algorithms.

We develop load balancing algorithms for WDM-based packet networks where the average traffic between nodes is dynamically changing. In WDM-based packet networks, routers are connected to each other using wavelengths (lightpaths) to form a logical network topology. The logical topology may be reconfigured by rearranging the lightpaths connecting the routers. Our algorithms reconfigure the logical topology to minimize the maximum link load.

Abstract—Load balancing problems for multiclass jobs in distributed/parallel computer systems with general network configurations are considered. We construct a general model of such a distributed/parallel computer system. The system consists of heterogeneous host computers/processors (nodes) which are interconnected by a generally configured communication/interconnection network wherein there are several classes of jobs, each of which has its distinct delay function at each host and each communication link. This model is used to formulate the multiclass job load balancing problem as a nonlinear optimization problem in which the goal is to minimize the mean response time of a job. A number of simple and intuitive theoretical results on the solution of the optimization problem are derived. On the basis of these results, we propose an effective load balancing algorithm for balancing the load over an entire distributed/parallel system. The proposed algorithm has two attractive features. One is that the algorithm can be implemented in a decentralized fashion. Another feature is simple and straightforward structure. Models of nodes, communication networks, and a numerical example are illustrated. The proposed algorithm is compared with a well-known standard steepest-descent algorithm, the FD algorithm. By using numerical experiments, we show that the proposed algorithm has much faster convergence in terms of computational time than the FD algorithm. Index Terms—Algorithms, communication networks, distributed/parallel computer systems, interconnection networks, load balancing, management, multiclass jobs, nonlinear optimization, performance, queuing theory. 1

There are many problems related to the design of networks. Among them, the message routing problem plays a determinant role in the optimization of network performance. Much of the motivation for this work comes from this problem which is shown to belong to the class of nonlinear convex multicommodity flow problems. This paper emphasizes the message routing problem in data networks, but it includes a broader literature overview of convex multicommodity flow problems. We present and discuss the main solution techniques proposed for solving this class of largescale convex optimization problems. We conduct some numerical experiments on the message routing problem with some different techniques. 1 Introduction The literature dealing with multicommodity flow problems is rich since the publication of the works of Ford and Fulkerson's [19] and T.C. Hu [30] in the beginning of the 1960s. These problems usually have a very large number of variables and constraints and arise in a great variety o...