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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 1994] and by [Aumann and Rabani 1998] 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 2009] that the gap is 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 undirected planar graphs by [Rao 99] (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.

Joint work with Ken-ichi Kawarabayashi, to appear in FOCS 2021.

https://ucdavis.zoom.us/j/92696386057?pwd=V3hrODBH...