Title: On constant multi-commodity flow-cut gaps for directed minor-free graphs,
Abstract: 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 \cite{linial1994geometry} and by Aumann and Rabani
\cite{aumann1998log} that for general $n$-vertex graphs it is bounded by
$O(\log n)$ and the Gupta-Newman-Rabinovich-Sinclair conjecture
\cite{gupta2004cuts} asserts that it is $O(1)$ for any family of graphs that
excludes some fixed minor.
The flow-cut gap is poorly understood for the case of directed graphs. We
show that for uniform demands it is $O(1)$ on directed series-parallel graphs,
and on directed graphs of bounded pathwidth. These are the first constant upper
bounds of this type for some non-trivial family of directed graphs. We also
obtain $O(1)$ upper bounds for the general multi-commodity flow-cut gap on
directed trees and cycles. These bounds are obtained via new embeddings and
Lipschitz quasipartitions for quasimetric spaces, which generalize analogous
results form the metric case, and could be of independent interest. Finally, we
discuss limitations of methods that were developed for undirected graphs, such
as random partitions, and random embeddings.