On the Edge Capacitated Steiner Tree Problem Arxiv:1607.07082V1
On the edge capacitated Steiner tree problem C´edricBentz(1), Marie-Christine Costa(2), Alain Hertz(3) (1) CEDRIC, CNAM, 292 rue Saint-Martin, 75003 Paris, France (2) ENSTA Paris-Tech, University of Paris-Saclay (and CEDRIC CNAM), 91762 Palaiseau Cedex, France (3) GERAD and D´epartement de math´ematiqueset g´enieindustriel, Polytechnique Montr´eal,Canada July 26, 2016 Abstract Given a graph G = (V; E) with a root r 2 V , positive capacities fc(e)je 2 Eg, and non-negative lengths f`(e)je 2 Eg, the minimum-length (rooted) edge capacitated Steiner tree problem is to find a tree in G of minimum total length, rooted at r, spanning a given subset T ⊂ V of vertices, and such that, for each e 2 E, there are at most c(e) paths, linking r to vertices in T , that contain e. We study the complexity and approximability of the problem, considering several relevant parameters such as the number of terminals, the edge lengths and the minimum and maximum edge capacities. For all but one combinations of assumptions regarding these parameters, we settle the question, giving a complete characterization that separates tractable cases from hard ones. The only remaining open case is proved to be equivalent to a long-standing open problem. We also prove close relations between our problem and classical Steiner tree as well as vertex-disjoint paths problems. 1 Introduction The graphs in this paper can be directed or undirected. Consider a connected graph G = (V; E) with a set T ⊂ V of terminal vertices, or simply terminals, and a length (or cost) function ` : E ! Q+.
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