On the single-source unsplittable flow problem

'Let G = (V, E) be a capacitated directed graph with a source s and k terminals t(i) with demands d(i), 1 less than or equal to i less than or equal to k. We would like to concurrently route every demand on a single path from s to the corresponding terminal without violating the capacities. There are several interesting and important Variations of this unsplittable flow problem. If the necessary cut condition is satisfied, we show how to compute an unsplittable flow satisfying the demands such that the total flow through ally edge exceeds its capacity by at most the maximum demand. For graphs in which all capacities are at least the maximum demand, we therefore obtain an unsplittable flow with congestion at most 2, and this result is best possible. Furthermore, we show that all demands can be routed unsplittably in 5 rounds, i.e., all demands call be collectively satisfied by the union of 5 unsplittable flows. Finally, we show that 22.6% of the total demand call be satisfied unsplittably. These results are extended to the case when the cut condition is not necessarily satisfied. We derive a a-approximation algorithm for congestion, a 5-approximation algorithm for the number of rounds and a 4.43 = 1/0.226-approximation algorithm for the maximum routable demand