hal-00676432 https://enpc.hal.science/hal-00676432 doi:10.1007/s10288-010-0150-8 [X] Ecole Polytechnique [ENPC] Ecole des Ponts ParisTech [UGA] HAL Grenoble Alpes [CNRS] CNRS - Centre national de la recherche scientifique [UNIV-GRENOBLE1] Université Joseph Fourier - Grenoble I [UNIV-MLV] Université de Marne la Vallée [INPG] Institut polytechnique de Grenoble [CERMICS] Centre d'Enseignement et de Recherche en Mathématiques, Informatique et Calcul Scientifique [G-SCOP] LABORATOIRE G-SCOP [INSMI] CNRS-INSMI - INstitut des Sciences Mathématiques et de leurs Interactions [X-DEP] Polytechnique [PARISTECH] ParisTech [G-SCOP_OSP_OC] Optimisation Combinatoire [G-SCOP_OSP] Optimisation des systèmes de Production [TDS-MACS] Réseau de recherche en Théorie des Systèmes Distribués, Modélisation, Analyse et Contrôle des Systèmes Solving coloring, minimum clique cover and kernel problems on arc intersection graphs of directed paths on a tree Durand de Gevigney, Olivier Meunier, Frédéric Popa, Christian Reygner, Julien Ayrin, Romero [MATH.MATH-OC] Mathematics [math]/Optimization and Control [math.OC] ART DE graph Directed tree Intersection graph Kernel Minimum clique cover Multicut Let T = (V, A) be a directed tree. Given a collection P of dipaths on T, we can look at the arc-intersection graph P whose vertex set is P and where two vertices are connected by an edge if the corresponding dipaths share a common arc. Monma and Wei, who started their study in a seminal paper on intersection graphs of paths on a tree, called them DE graphs (for directed edge path graphs) and proved that they are perfect. DE graphs find one of their applications in the context of optical networks. For instance, assigning wavelengths to set of dipaths in a directed tree network consists in finding a proper coloring of the arc-intersection graph. In the present paper, we give - a simple algorithm finding a minimum proper coloring of the paths. - a faster algorithm than previously known ones finding a minimum multicut on a directed tree. It runs in O({pipe}V{pipe}{pipe}P{pipe}) (it corresponds to the minimum clique cover of I (P, T)). - a polynomial algorithm computing a kernel in any DE graph whose edges are oriented in a clique-acyclic way. Even if we know by a theorem of Boros and Gurvich that such a kernel exists for any perfect graph, it is in general not known whether there is a polynomial algorithm (polynomial algorithms computing kernels are known only for few classes of perfect graphs). 2011 en 4OR: A Quarterly Journal of Operations Research Springer Verlag