Cost-distance: two metric network design

Presents the cost-distance problem, which consists of finding a Steiner tree which optimizes the sum of edge costs along one metric and the sum of source-sink distances along an unrelated second metric. We give the first known O(log k) randomized approximation scheme for the cost-distance problem, where k is the number of sources. We reduce several common network design problems to cost-distance problems, obtaining (in some cases) the first known logarithmic approximation for them. These problems include a single-sink buy-at-bulk problem with variable pipe types between different sets of nodes, facility location with buy-at-bulk-type costs on edges, constructing single-source multicast trees with good cost and delay properties, and multi-level facility location. Our algorithm is also easier to implement and significantly faster than previously known algorithms for buy-at-bulk design problems.