Approximating the single source unsplittable min-cost flow problem

In the single source unsplittable min-cost flow problem, commodities must be routed simultaneously from a common source vertex to certain destination vertices in a given graph with edge capacities and costs; the demand of each commodity must be routed along a single path and the total cost must not exceed a given budget. This problem has been introduced by J.M. Kleinberg (1996) and generalizes several NP-complete problems from various areas in combinatorial optimization such as packing, partitioning, scheduling load balancing, and virtual-circuit routing. S.G. Kolliopoulos and C. Stein (2000) and Y.N. Dinitz et al. (1999) developed algorithms improving the first approximation results of Kleinberg for the problem to minimize the violation of edge capacities and for other variants. However, none of the developed techniques is capable of providing solutions without also violating the cost constraint. We give the first approximation results with hard cost constraints. Moreover all our results dominate the best known bicriteria approximations. Finally, we provide results on the hardness of approximation for several variants of the problem.