Convex Combinations of Single Source Unsplittable Flows

In the single source unsplittable flow problem, commodities must be routed simultaneously from a common source vertex to certain destination vertices in a given digraph. The demand of each commodity must be routed along a single path. In a ground breaking paper Dinitz, Garg, and Goemans [4] prove that any given (splittable) flow satisfying certain demands can be turned into an unsplittable flow with the following nice property: In the unsplittable flow, the flow value on any arc exceeds the flow value on that arc in the given flow by no more than the maximum demand. Goemans conjectures that this result even holds in the more general context with arbitrary costs on the arcs when it is required that the cost of the unsplit-table flow must not exceed the cost of the given (splittable) flow. The following is an equivalent formulation of Goemans' conjecture: Any (splittable) flow can be written as a convex combination of unsplittable flows such that the unsplittable flows have the nice property mentioned above. We prove a slightly weaker version of this conjecture where each individual unsplittable flow occurring in the convex combination does not necessarily fulfill the original demands but rounded demands. Preliminary computational results based on our underlying algorithm support the strong version of the conjecture.