Due to the success of the Internet and the diversity of communication applications, it is becoming increasingly difficult to forecast traffic patterns. To capture the traffic variations, we introduce a flexible model where traffic belongs to a polytope. We assume that the traffic demands between nodes can be carried on many paths, with respect to network resources. Moreover, to guarantee the network stability and to make the routing easy to implement, the proportions of traffic flowing through each path have to be independent of the current traffic demands. We show that a minimum-cost routing satisfying the previous properties can be efficiently computed by column and constraint generations. We then present several strategies related to certain algorithmic details. Finally, theoretical and computational studies show that this new flexible model can be much more economical than a classical deterministic model based on a given traffic matrix. This paper can be considered as a mathematical framework for a new flexible virtual private network service offer. It also introduces a new concept: the routing of a polytope.
Most of integer, convex, and large-scale linear problems are solved using cutting plane and column generation algorithms. Therefore, to handle large-size problems and to reduce the computing times, it may be very useful to accelerate cutting plane algorithms. We show in this article that we can achieve this goal by choosing good separation points. Focus is given on problems for which we have an exact separation oracle. An in-out algorithm is proposed, and the convergence is proved under some general assumptions. Computational experiments related to three classes of problems, survivable network design, multicommodity flow problems, and random linear programs, clearly point out the savings of time allowed by the simple in-out approach proposed in this article.
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