On constant multi-commodity flow-cut gaps for directed minor-free graphs

TL;DR

This paper establishes the first constant upper bounds for the multi-commodity flow-cut gap in certain classes of directed graphs, advancing understanding of flow-cut relationships beyond undirected graphs.

Contribution

It proves that the flow-cut gap is constant for directed series-parallel graphs, graphs of bounded pathwidth, directed trees, and cycles, using new embeddings and quasipartitions.

Findings

Flow-cut gap is O(1) for directed series-parallel graphs.

Flow-cut gap is O(1) for directed graphs of bounded pathwidth.

New embeddings and Lipschitz quasipartitions are developed for quasimetric spaces.

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.