A Simple Greedy Algorithm for the k-Disjoint Flow Problem

In classical network flow theory the choice of paths, on which flow is sent, is only restricted by arc capacities. This, however, is not realistic in most applications. Many problems restrict, e.g., the number of paths being used to route a commodity. One idea to increase reliability of routings, e.g., in telecommunication, is to copy a demand and send the copies along disjoint paths. Such problems theoretically result in the

k

-disjoint flow problem (

k

-DFP). This problem is a variant of the classical multicommodity flow problem with the additional requirement that the number of paths to route a commodity is bounded by a given parameter. Moreover, all paths used by the same commodity have to be arc disjoint.

We present a simple greedy algorithm for the optimization version of the

k

-DFP where the objective is to maximize the sum of routed demands. This algorithm generalizes a greedy algorithm by Kolman and Scheideler (2002) that approximates the corresponding unsplittable flow problem, in which every commodity may be routed along a single path only. We achieve an approximation factor of

$O(k_{\text{max}} \sqrt{m}/k_{\text{min}})$

, where

m

is the number of arcs and

$k_{\text{max}}$

(

$k_{\text{min}}$

) is the maximum (minimum) bound on the number of paths allowed to route any of the commodities. We argue that this performance guarantee is best possible for instances where

$k_{\text{max}}/k_{\text{min}}$

is constant, unless

$\mathcal{P}=\mathcal{P}\mathcal{N}$

.