Interchanging distance and capacity in probabilistic mappings

Harald Räcke [STOC 2008] described a new method to obtain hierarchical decompositions of networks in a way that minimizes the congestion. Räcke’s approach is based on an equivalence that he discovered between minimizing congestion and minimizing stretch (in a certain setting). Here we present Räcke’s equivalence in an abstract setting that is more general than the one described in Räcke’s work, and clarifies the power of Räcke’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