On the approximation of the generalized capacitated tree-routing problem

In this paper, we study the generalized capacitated tree-routing problem (GCTR), which was introduced to unify the several known multicast problems in networks with edge/demand capacities. Let G = (V, E) be a connected underlying graph with a bulk edge capacity lambda > 0 and an edge weight w(e) >= 0, e is an element of E; we are allowed to construct a network on G by installing any edge capacity k(e)lambda with an integer ke >= 0 for each edge e is an element of E, where the resulting network costs Sigma e is an element of (E) k(e)w(e). Given a sink s is an element of V, a set M subset of V of terminals with a demand q(V) >= 0, V is an element of M and a demand capacity. > 0, we wish to construct the minimum cost network so that all the demands can be sent to s along a suitable collection T = {T1, T2,..., T p} of trees rooted at s, where the total demand collected by each tree Ti is bounded from above by., and the flow amount f (e) of T that goes through each edge e is bounded from above by the edge capacity ke lambda. In this paper, f (e) is defined as Sigma Ti is an element of T: e is an element of Ti-[ (alpha +) (beta qTi (e)]) for prescribed constants alpha, beta >= 0, where qTi (e) denotes the total demand that passes through the edge e along Ti. The term a means a fixed amount used to establish the routing Ti by separating the inside of Ti from the outside while the term alpha qTi (e) means the net capacity proportional to the demand qTi (e). The objective of GCTR is to construct a minimum cost network that admits a collection T of trees to send all demand to sink. It was left open to show whether GCTR with lambda< alpha+ beta k. is approximable by a constant factor or not. In this paper, we present a 13.037-approximation algorithm to GCTR for this case. (C) 2009 Elsevier B. V. All rights reserved.