An Approximate Generalization of the Okamura-Seymour Theorem

We consider the problem of multi-commodity flows in planar graphs. Okamura and Seymour showed that if all the demands are incident on one face, then the cut-condition is sufficient for routing demands. We consider the following generalization of this setting and prove an approximate max flow-min cut theorem: for every demand edge, there exists a face containing both its end points. We show that the cut-condition is sufficient for routing $\Omega(1)$-fraction of all the demands. To prove this, we give a $L_{1}$-embedding of the planar metric which approximately preserves distance between all pair of points on the same face.