Quasimetric embeddings and their applications

TL;DR

This paper extends metric embedding techniques to quasimetric spaces, especially those from directed graphs, and applies these results to improve approximation algorithms for graph partitioning problems.

Contribution

It introduces random quasipartitions for quasimetric spaces and provides bounds on embeddings into quasiultrametrics, with applications to graph algorithms.

Findings

Embedding quasimetric spaces into quasiultrametrics with low distortion is possible for spaces supported on graphs with bounded treewidth.

New approximation algorithms for Directed Non-Bipartite Sparsest-Cut and Multicut problems with polynomial runtime.

Lower bounds on embedding distortions highlight fundamental differences between metric and quasimetric spaces.

Abstract

We study generalizations of classical metric embedding results to the case of quasimetric spaces; that is, spaces that do not necessarily satisfy symmetry. Quasimetric spaces arise naturally from the shortest-path distances on directed graphs. Perhaps surprisingly, very little is known about low-distortion embeddings for quasimetric spaces. Random embeddings into ultrametric spaces are arguably one of the most successful geometric tools in the context of algorithm design. We extend this to the quasimetric case as follows. We show that any $n$-point quasimetric space supported on a graph of treewidth $t$ admits a random embedding into quasiultrametric spaces with distortion $O(t \log^2 n)$, where quasiultrametrics are a natural generalization of ultrametrics. This result allows us to obtain $t\log^{O(1)} n$-approximation algorithms for the Directed Non-Bipartite Sparsest-Cut and the Directed Multicut problems on $n$-vertex graphs of treewidth $t$, with running time polynomial in both $n$ and $t$. The above results are obtained by considering a generalization of random partitions to the quasimetric case, which we refer to as random quasipartitions. Using this definition and a construction of [Chuzhoy and Khanna 2009] we derive a polynomial lower bound on the distortion of random embeddings of general quasimetric spaces into quasiultrametric spaces. Finally, we establish a lower bound for embedding the shortest-path quasimetric of a graph $G$ into graphs that exclude $G$ as a minor. This lower bound is used to show that several embedding results from the metric case do not have natural analogues in the quasimetric setting.