  The multi-commodity flow-cut gap is a fundamental parameter that affects the
performance of several divide \& conquer algorithms, and has been extensively
studied for various classes of undirected graphs. It has been shown by Linial,
London and Rabinovich and by Aumann and Rabani that for general $n$-vertex
graphs it is bounded by $O(\log n)$ and the Gupta-Newman-Rabinovich-Sinclair
conjecture asserts that it is $O(1)$ for any family of graphs that excludes
some fixed minor.
  We show that the multicommodity flow-cut gap on \emph{directed} planar graphs
is $O(\log^3 n)$. This is the first \emph{sub-polynomial} bound for any family
of directed graphs of super-constant treewidth. We remark that for general
directed graphs, it has been shown by Chuzhoy and Khanna that the gap is
$\widetilde{\Omega}(n^{1/7})$, even for directed acyclic graphs.
  As a direct consequence of our result, we also obtain the first
polynomial-time polylogarithmic-approximation algorithms for the Directed
Non-Bipartite Sparsest-Cut, and the Directed Multicut problems for directed
planar graphs, which extends the long-standing result for undirectd planar
graphs by Rao (with a slightly weaker bound).
  At the heart of our result we investigate low-distortion quasimetric
embeddings into \emph{directed} $\ell_1$. More precisely, we construct
$O(\log^2 n)$-Lipschitz quasipartitions for the shortest-path quasimetric
spaces of planar digraphs, which generalize the notion of Lipschitz partitions
from the theory of metric embeddings. This construction combines ideas from the
theory of bi-Lipschitz embeddings, with tools form data structures on directed
planar graphs.
