On Network Design Problems: Fixed Cost Flows and the Covering Steiner Problem

Network design problems, such as generalizations of the Steiner Tree Problem, can be cast as edge-cost-flow problems. An edge -cost flow problem is a min-cost flow problem in which the cost of the flow equals the sum of the costs of the edges carrying positive flow. We prove a hardness result for the Minimum Edge Cost Flow Problem (MECF). Using the oneround two -prover scenario, we prove that MECF does not admit a 2(log1-epsilon n)-ratio approximation, for every constant epsilon > 0, unless NP subset of DTIME(n(polylogn)) A restricted version of MECF, called Infinite Capacity MECF (ICF), is defined. The ICF problem is defined as follows: (i) all edges have infinite capacity, (ii) there are multiple sources and sinks, where flow can be delivered from every source to every sink, (iii) each source and sink has a supply amount and demand amount, respectively, and (iv) the required total flow is given as part of the input. The goal is to find a minimum edge-cost flow that meets the required total flow while obeying the demands of the sinks and the supplies of the sources. This problem naturally arises in practical scheduling applications, and is equivalent to the special case of single source MECF, with all edges not touching the source or the sink having infinite capacity. The directed ICF generalizes the Covering Steiner Problem in directed and undirected graphs. The undirected version of ICF generalizes several network design problems, such as: Steiner Tree Problem, k -MST, Point-to-point Connection Problem, and the generalized Steiner Tree Problem. An O (log x) -approximation algorithm for undirected ICF is presented. We also present a bi-criteria approximation algorithm for directed ICF. The algorithm for directed ICF finds a flow that delivers half the required flow at a cost that is at most O (n(epsilon)/epsilon(4)) times bigger than the cost of an optimal flow. The running time of the algorithm is O (x(2/epsilon) . n(1+1/epsilon)), where x denotes the required total flow. Randomized approximation algorithms for the Covering Steiner Problem in directed and undirected graphs are presented. The algorithms are based on a randomized reduction to a problem called 1/2-Group Steiner. In undirected graphs, the approximation ratio matches the approximation ratio of Konjevod et al. [2002]. However, our algorithm is much simpler. In directed graphs, the algorithm is the first nontrivial approximation algorithm for the Covering Steiner Problem. Deterministic algorithms are obtained by derandomization.