Flows on Few Paths: Algorithms and Lower Bounds

In classical network flow theory, flow being sent from a source to a destination may be split into a large number of chunks traveling on different paths through the network. This effect is undesired or even forbidden in many applications. Kleinberg introduced the unsplittable flow problem where all flow traveling from a source to a destination must be sent on only one path. This is a generalization of the NP-complete edgedisjoint paths problem. In particular, the randomized rounding technique of Raghavan and Thompson can be applied. A generalization of unsplittable flows are k-splittable flows where the number of paths used by a commodity i is bounded by a given integer ki. The contribution of this paper is twofold. First, for the unsplittable flow problem, we prove a lower bound of Ω(logm/ log logm) on the performance of randomized rounding. This result almost matches the best known upper bound of O(logm). To the best of our knowledge, the problem of finding a non-trivial lower bound has so far been open. In the second part of the paper, we study a new variant of the k-splittable flow problem with additional constraints on the amount of flow being sent along each path. The motivation for these constraints comes from the following packing and routing problem: A commodity must be shipped using a given number of containers of given sizes. First, one has to make a decision on the fraction of the commodity packed into each container. Then, the containers must be routed through a network whose edges correspond, for example, to ships or trains. Each edge has a capacity bounding the total size or weight of containers which are being routed on it. We present approximation results for two versions of this problem with multiple commodities and the objective to minimize the congestion of the network. The key idea is to reduce the problem under consideration to an unsplittable flow problem while only losing a constant factor in the performance ratio.