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We consider the problem of multicommodity flows in planar graphs. Seymourshowed that if the union of supply and demand graphs is planar, then the cutcondition is sufficient for routing demands. Okamura-Seymour showed that if alldemands are incident on one face, then again cut condition is sufficient forrouting demands. We consider a common generalization of these settings wherethe end points of each demand are on the same face of the planar graph. We showthat if the source sink pairs on each face of the graph are such that sourcesand sinks appear contiguously on the cycle bounding the face, then the flow cutgap is at most 3. We come up with a notion of approximating demands on a faceby convex combination of laminar demands to prove this result.

Multicommodity Flows in Planar Graphs with Demands on Faces