  The classical Okamura-Seymour theorem states that for an edge-capacitated,
multi-commodity flow instance in which all terminals lie on a single face of a
planar graph, there exists a feasible concurrent flow if and only if the cut
conditions are satisfied. Simple examples show that a similar theorem is
impossible in the node-capacitated setting. Nevertheless, we prove that an
approximate flow/cut theorem does hold: For some universal c > 0, if the node
cut conditions are satisfied, then one can simultaneously route a c-fraction of
all the demands. This answers an open question of Chekuri and Kawarabayashi.
More generally, we show that this holds in the setting of multi-commodity
polymatroid networks introduced by Chekuri, et. al. Our approach employs a new
type of random metric embedding in order to round the convex programs
corresponding to these more general flow problems.
