  The single-source unsplittable flow (SSUF) problem asks to send flow from a
common source to different terminals with unrelated demands, each terminal
being served through a single path. One of the most heavily studied SSUF
objectives is to minimize the violation of some given arc capacities. A seminal
result of Dinitz, Garg, and Goemans showed that, whenever a fractional flow
exists respecting the capacities, then there is an unsplittable one violating
the capacities by at most the maximum demand. Goemans conjectured a very
natural cost version of the same result, where the unsplittable flow is
required to be no more expensive than the fractional one. This intriguing
conjecture remains open. More so, there are arguably no non-trivial graph
classes for which it is known to hold.
  We show that a slight weakening of it (with at most twice as large
violations) holds for planar graphs. Our result is based on a connection to a
highly structured discrepancy problem, whose repeated resolution allows us to
successively reduce the number of paths used for each terminal, until we obtain
an unsplittable flow. Moreover, our techniques also extend to simultaneous
upper and lower bounds on the flow values. This also affirmatively answers a
conjecture of Morell and Skutella for planar SSUF.
