Uniform multicommodity flow in the hypercube with random edge capacities

TL;DR

This paper analyzes multicommodity flows in high-dimensional hypercubes with random edge capacities, showing that the network can support near-maximum flows between all vertex pairs with high probability as the dimension grows.

Contribution

It provides the first probabilistic bounds on multicommodity flow capacities in hypercubes with random capacities, establishing near-optimal flow support results.

Findings

Supports flows close to expected capacity between antipodal pairs

Supports flows close to scaled expected capacity between all pairs

Results are asymptotically optimal as dimension increases

Abstract

We give two results for multicommodity flows in the $d$-dimensional hypercube ${Q}^d$ with independent random edge capacities distributed like $C$ where $\Pr[C>0]>1/2$. Firstly, with high probability as $d \rightarrow \infty$, the network can support simultaneous multicommodity flows of volume close to $E[C]$ between all antipodal vertex pairs. Secondly, with high probability, the network can support simultaneous multicommodity flows of volume close to $2^{1-d} E[C]$ between all vertex pairs. Both results are best possible.