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 @l>0 and an edge weight w(e)>=0, e@?E; we are allowed to construct a network on G by installing any edge capacity k"e@l with an integer k"e>=0 for each edge e@?E, where the resulting network costs @?"e"@?"Ek"ew(e). Given a sink s@?V, a set M@?V of terminals with a demand q(v)>=0, v@?M, and a demand capacity @k>0, we wish to construct the minimum cost network so that all the demands can be sent to s along a suitable collection T={T"1,T"2,...,T"p} of trees rooted at s, where the total demand collected by each tree T"i is bounded from above by @k, and the flow amount f(e) of T that goes through each edge e is bounded from above by the edge capacity k"e@l. In this paper, f(e) is defined as @?"T"""i"@?"T":"e"@?"T"""i[@a+@bq"T"""i(e)] for prescribed constants @a,@b>=0, where q"T"""i(e) denotes the total demand that passes through the edge e along T"i. The term @a means a fixed amount used to establish the routing T"i by separating the inside of T"i from the outside while the term @bq"T"""i(e) means the net capacity proportional to the demand q"T"""i(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 @l<@a+@b@k is approximable by a constant factor or not. In this paper, we present a 13.037-approximation algorithm to GCTR for this case.