Approximating the Single-Sink Link-Installation Problem in Network Design

We initiate the algorithmic study of an important but NP hard problem that arises commonly in network design The input consists of An undirected graph with one sink node and multiple source nodes a speci ed length for each edge and a speci ed demand demv for each source node v A small set of cable types where each cable type is speci ed by its capacity and its cost per unit length The cost per unit capacity per unit length of a high capacity cable may be signi cantly less than that of a low capacity cable re ecting an economy of scale i e the payo for buying at bulk may be very high The goal is to design a minimum cost network that can simultaneously route all the demands at the sources to the sink by installing zero or more copies of each cable type on each edge of the graph An additional restriction is that the demand of each source must follow a single path The problem is to nd a route from each source node to the sink and to assign capacity to each edge of the network such that the total costs of cables installed is minimized We call this problem the single sink link installation problem For the general problem we introduce a new moat type lower bound on the optimal value and we prove a useful structural property of near optimal solutions For every instance of our problem there is a near optimal solution whose graph is acyclic with cost no more than twice the optimal cost We present e cient approximation algorithms for key special cases of the problem that arise in practice For points in the Euclidean plane we give an approximation algorithm with performance guarantee O log D u where D is the total demand and u is the smallest cable capacity When the metric is arbitrary we consider the case where the network to be designed is restricted to be two level i e every source sink path has at most two edges For this problem we present an algorithm with performance guarantee O logn where n is the number of nodes in the input graph and also show that this performance guarantee is nearly best possible