Approximating the Single-Sink Edge 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 (1) An undirected graph with one sink node and multiple source nodes, a speciied length for each edge, and a speciied demand, dem v , for each source node v. (2) A small set of cable types, where each cable type is speciied by its capacity and its cost per unit length. The cost per unit capacity per unit length of a high-capacity cable may be signiicantly less than that of a low-capacity cable, reeecting an economy of scale, i.e., the payoo 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. Thus, the problem is to nd a route for each sink node 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 edge-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 eecient 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), where D is the total demand. 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(log n), where n is the number of nodes in the input graph.