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Let G = (V, E) be a capacitated directed graph with a source s and k terminals t i with demands d i , 1≤ i ≤ 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 any 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 can be collectively satisfied by the union of 5 unsplittable flows. Finally, we show that 22.6% of the total demand can be satisfied unsplittably.These results are extended to the case when the cut condition is not necessarily satisfied. We derive a 2-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.

The problem of finding a minimum weight k-vertex connected spanning subgraph in a graph G = (V, E) is considered. For k ≥ 2, this problem is known to be NP-hard. Combining properties of inclusionminimal k-vertex connected graphs and of k-out-connected graphs (i.e., graphs which contain a vertex from which there exist k internally vertex-disjoint paths to every other vertex), we derive an auxiliary polynomial time algorithm for finding a ( k 2 + 1)-connected subgraph with a weight at most twice the optimum to the original problem. In particular, we obtain a 2-approximation algorithm for the case k = 3 of our problem. This improves the best previously known approximation ratio 3. The complexity of the algorithm is O(|V | 3 |E|) = O(|V | 5 ). * Up to 1990,