An Approximate Generalization of the Okamura-Seymour Theorem

08/01/2022

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by   Nikhil Kumar, et al.

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We consider the problem of multicommodity 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 Ω(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.