Maximizing the number of requests in oriented trees with a grooming factor

<div> The Maximum All Request Path Grooming (MARPG) problem consists in finding the maximum number of requests (connections) which can be established in a network, where each arc has a given capacity or bandwidth C (grooming factor). The Maximum Path Coloring problem consists for a given number of colors (wavelengths) W in finding the maximum number of requests that can be established so that two requests sharing an arc have different colors. These problems are part of the more general RWA (Routing and Wavelength Assignment) problem and have been studied for various classes of networks like paths, dipaths, undirected trees and symmetric directed trees. Here we consider the case where the network is an oriented tree (tree in which each edge has a unique orientation) where the two problems are equivalent. We give the value of the maximum number of requests for various families of oriented trees like Fig-Trees. To do that we revisit the problem when the network is a directed path by giving the structure of a maximum set of requests and determining bounds on the maximum load of an arc of the dipath. These bounds can be used for computing the cutwidth of a graph. </div>