Directed paths on a tree: coloring, multicut and kernel

In a paper published in Journal of Combinatorial Theory, Series B (1986), Monma and Wei propose an extensive study of the intersection graph of paths on a tree. They explore this notion by varying the notion of intersection: the paths are respectively considered to be the sets of their vertices and the sets of their edges, and the trees may or may not be directed. Their main results are a characterization of these graphs in term of their clique tree and a unified recognition algorithm. Related optimization problems are also presented. In the present note, we are interested by the case when the tree is directed, and the paths are considered to be the set of their arcs. An application is the question of wavelength assignment in optimal network. In their article, Monma and Wei show that in this case the intersection graph is a perfect graph and they give polynomial combinatorial algorithms to solve the minimum coloring and the maximum stable problems, with the help of the clique tree representation (Tarjan’s method). The maximum clique problem is immediate. They leave the problem of finding a minimum clique cover, which is here nothing else than the minimum multicut of the dipaths, as an open question. Costa, Létocart and Roupin in a survey about multicut and integer multiflow noted that the maximum stable, which is a maximum set of arc-disjoint dipaths, and the minimum clique cover, which is the minimum set of arcs intersecting all dipaths (minimum multicut), are integers solutions of a linear program with totally unimodular matrices, and hence can be solved polynomially. For the maximum stable problem (or maximum set of arc-disjoint dipaths), there is a simple polynomial algorithm found by Garg, Vazirani and Yannakakis. In the present paper, we present faster algorithms for solving the minimum coloring and the minimum clique cover problems (minimum multicut), and maybe simpler, since we work directly with the dipaths on the tree and avoid both the clique decomposition and the linear programming formulation. They both run in O(np) time, where n is the number of vertices of the tree and p the number of paths. Another result is a polynomial algorithm computing a kernel in the intersection graph, when its edges are oriented in a clique-acyclic way. Indeed, such a kernel exists for any perfect graph by a theorem of Boros and Gurvich. Such algorithms computing kernels are known only for few classes of perfect graphs.