Interchanging distance and capacity in probabilistic mappings

TL;DR

This paper generalizes Harald Racke's equivalence between minimizing congestion and stretch in network decompositions, extending its applicability beyond specific settings and clarifying its theoretical power, while also discussing a related equivalence for planar graphs.

Contribution

It presents a more general abstract framework for Racke's equivalence and introduces a related equivalence applicable to planar graphs, enhancing understanding of network decompositions.

Findings

Generalized Racke's equivalence in an abstract setting

Clarified the theoretical power of the equivalence

Discussed a planar graph-specific equivalence

Abstract

Harald Racke [STOC 2008] described a new method to obtain hierarchical decompositions of networks in a way that minimizes the congestion. Racke's approach is based on an equivalence that he discovered between minimizing congestion and minimizing stretch (in a certain setting). Here we present Racke's equivalence in an abstract setting that is more general than the one described in Racke's work, and clarifies the power of Racke's result. In addition, we present a related (but different) equivalence that was developed by Yuval Emek [ESA 2009] and is only known to apply to planar graphs.