On the Single-Source Unsplittable Flow Problem

Abstract

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 k-Splittable Flow Problem

“…The randomized rounding technique of Raghavan and Thompson [17] yields a logarithmic approximation. For the case of a single-source node, there even exist constant factor approximation algorithms, see [9], [13], [6], and [18]. More details can be found in the Introduction.…”

“…Hence, any ρ-approximation algorithm for the unsplittable flow problem which compares the approximate solution with a solution to the fractional relaxation already leads to a ρ-approximation algorithm for the k-splittable flow problem. In particular, this applies to the currently best known approximation algorithms for the maximum concurrent unsplittable flow problem [8], [6], [18].…”

“…If the balance condition is not given, they can achieve a performance guarantee of 3 + 2 √ 2 for the case without costs. Dinitz et al [6] consider the problem without costs. They give a 2-approximation for the case with balance condition, and a 5-approximation, otherwise.…”

Untitled

“…Another direction of research focuses on flows under the restriction that each commodity is allowed to use only a limited number of paths: the edge disjoint paths problem and the unsplittable flow problem allow one path per commodity [8,10,11,19,23,25,27,26,34]; the h-splittable flow problem allows at most h, not necessarily disjoint, paths per commodity [7,24,31,30]; particular attention has been given to single source unsplittable flow problems [13,16,25,33]. Though there is a certain similarity between the h-splittable flows and the h-route flows (in fact, they may even coincide for some instances), there is also a substantial difference.…”

Untitled

“…Another direction of research focuses on flows under the restriction that each commodity is allowed to use only a limited number of paths: the edge disjoint paths problem and the unsplittable flow problem allow one path per commodity [8,10,11,19,23,25,27,26,34]; the h-splittable flow problem allows at most h, not necessarily disjoint, paths per commodity [7,24,31,30]; particular attention has been given to single source unsplittable flow problems [13,16,25,33]. Though there is a certain similarity between the h-splittable flows and the h-route flows (in fact, they may even coincide for some instances), there is also a substantial difference.…”