Abstract. In this survey the following model is considered. We assume that an instance I of a computationally hard optimization problem has been solved and that we know the optimum solution of such instance. Then a new instance I ′ is proposed, obtained by means of a slight perturbation of instance I. How can we exploit the knowledge we have on the solution of instance I to compute a (approximate) solution of instance I ′ in an efficient way? This computation model is called reoptimization and is of practical interest in various circumstances. In this article we first discuss what kind of performance we can expect for specific classes of problems and then we present some classical optimization problems (i.e. Max Knapsack, Min Steiner Tree, Scheduling) in which this approach has been fruitfully applied. Subsequently, we address vehicle routing problems and we show how the reoptimization approach can be used to obtain good approximate solution in an efficient way for some of these problems.

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We consider the on-line channel assignment problem in the case of cellular networks and we formalize this problem as an on-line load balancing problem for temporary tasks with restricted assignment. For the latter problem, we provide a general solution (denoted as the cluster algorithm) and we characterize its competitive ratio in terms of the combinatorial properties of the graph representing the network. We then compare the cluster algorithm with the greedy one when applied to the channel assignment problem: it turns out that the competitive ratio of the cluster algorithm is strictly better than the competitive ratio of the greedy algorithm. The cluster method is general enough to be applied to other on-line load balancing problems and, for some topologies, it can be proved to be optimal. (C) 2003 Elsevier B.V. All rights reserved

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