Convex Combinations of Single Source Unsplittable Flows

Abstract

Abstract. 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 groundbreaking 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 unsplittable 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.

An Improved Upper Bound for the Ring Loading Problem

“…In a broader context, the Ring Loading Problem is a special case of unsplittable multicommodity flows. We mention the case of single source unsplittable flows, as similarities between theorems and conjectures for these problems exist (see [DGG99,Sku02,GC07,MSS07]). We also refer to the survey of Shepherd [She09].…”

Optimization in Wireless Networks

“…The wUMCFC problem, considered in this article, generalizes the UMCF problem with introducing additional coding and interference constraints. wUMCFC is proved to be N P−hard in [42] by a reduction to the single-source unsplittable flow problem [19,34,44].…”

Branch‐and‐price for energy optimization in multi‐hop wireless sensor networks

“…The wUMCFC problem is formulated as an UMCF problem with additional network coding and clique capacity constraints. The wUMCFC problem can be reduced to a single‐source unsplittable flow problem [23, 36, 37, 43] by considering a single source demand, where clique sets are singletons, and the coding variables are fixed to zero.…”

“…The wUMCFC problem can be reduced to a single-source unsplittable flow problem [23,36,37,43] by considering a single source demand, where clique sets are singletons, and the coding variables are fixed to zero. The single-source unsplittable flow problem is  -hard, hence the wUMCFC problem is also  -hard.…”