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

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 [15] and by Aumann and Rabani [3] that for general n-vertex graphs it is bounded by Oplog nq and the GuptaNewman-Rabinovich-Sinclair conjecture [9] asserts that it is Op1q 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 Op1q 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 Op1q 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. ∗Dept. of Computer Science & Engineering, The Ohio State University, [email protected]. †Dept. of Computer Science, University of Illinois at Chicago, [email protected]. ‡Dept. of Computer Science & Engineering, The Ohio State University, [email protected]. ar X iv :1 71 1. 01 37 0v 1 [ cs .D S] 4 N ov 2 01 7