Maximum Flows on Disjoint Paths

We consider the question: What is the maximum flow achievable in a network if the flow must be decomposable into a collection of edgedisjoint paths? Equivalently, we wish to find a maximum weighted packing of disjoint paths, where the weight of a path is the minimum capacity of an edge on the path. Our main result is an Ω(log n) lower bound on the approximability of the problem. We also show this bound is tight to within a constant factor. Surprisingly, the lower bound applies even for the simple case of undirected, planar graphs. Our results extend to the case in which the flow must decompose into at most k disjoint paths. There we obtain Θ(log k) upper and lower approximability bounds.