1 SOLVING THE OPTIMAL ROUTING PROBLEM IN A PACKET-SWITCHED COMPUTER NETWORK USING DECOMPOSITION Ahmed A. A. Radwan 1 Enas F. El-Geldawi 2 Department of Computer science, Faculty of Science 1 E-mail: aaaradwaneg@yahoo.co.uk 2 E-mail: enas.elgeldawi@mu.edu.eg Abstract: The message routing problem plays a determinant role in the optimization of network performance. Much of the motivation of this work comes from this problem which is shown to belong to the class of nonlinear convex multicommodity flow problems The flow deviation method is a well-known approach to solve this problem although its convergence rate near the optimal tends to be very slow. In this paper we discuss different techniques for improving the convergence rate of this method to solve the considered problem. We conduct some numerical experiments on the message routing problem with these techniques. Key Words: Computer networks, minimum cost flow, multicommodity flow, packet-switching. 1. Introduction A packet-switched computer network can be modeled as a graph in which many nodes are connected through a number of links. The purpose of the routing algorithm is to guide data packets from the nodes, where the packets originate, to their destinations in such a way as to minimize a given cost function. Adaptive single path algorithms, such as the adaptive shortest path algorithm, could oscillate between a collection of paths, which could be very heavy [2]. Multipath routing allows minimizing the average delay (cost function). Such optimal multipath routing algorithms avoid oscillation, if there is no change in traffic load and network topology. The solution of the optimal routing problem can be found using iterative algorithms such as the flow deviation method. This method is a special case of the so-called Frank-Wolfe method for solving general nonlinear programming problems with convex constraint sets. Our major interest in this paper is to improve the efficiency of the flow deviation method used to solve the optimal routing problem in Packet-Switched computer networks using the decomposition technique. This technique has been used by Radwan [11] for solving the traffic assignment problem and it has been found that the decomposed technique gives much better results than the undecomposed one (at least for the small size to medium size networks). This paper is organized as follows. In section 2, we will develop our network model and introduce the problem formulation to be solved. A solution procedure for solving the problem using the flow deviation method will be described in section 3. In section 4, we introduce the decomposition of the flow deviation method. In section 5 we introduce a modification of the flow deviation using PARTAN direction. In section 6, we combine the decomposed flow deviation method with PARTAN. The test cases are given in section 7. Finally, results are given in section 8. 2. Formulation of the model A computer network can be represented by a directed graph G = (V,A), where V = {1,2,...,n} is the set of nodes and A represents the set of links where link (i, j) is directed from node i to node j. Let W be the set of ordered node pairs referred to as origin–destination (OD) pairs. For each OD pair W, the input traffic arrival process (measured in data unit/sec) is assumed stationary with rate r . The routing objective is to divide each r  among the many paths, and hence links, from origin to destination in a way that the resulting total link flow pattern minimizes the average time delay [2]. That is equivalent to minimizing the function