The multicommodity flow problem is NP-hard already for two commodities over bipartite graphs. Nonetheless, using our recent theory of n-fold integer programming and extensions developed herein, we are able to establish the surprising polynomial time solvability of the problem in two broad situations.
@article{arxiv.0906.5106,
title = {Multicommodity Flow in Polynomial Time},
author = {Raymond Hemmecke and Shmuel Onn and Robert Weismantel},
journal= {arXiv preprint arXiv:0906.5106},
year = {2011}
}