Embeddings of Planar Quasimetrics into Directed \ell_1$ and Polylogarithmic Approximation for Directed Sparsest-Cut

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.