In the Maximum Multi-Commodity Flow Problem we have a transportation network and a finite number of different commodities. Each commodity i must be transported through the network from its production site s_i to its destination t_i . All goods are measured in the same unit (say, containers). Each link in the network can only transport goods in one direction and has a limited capacity as to how much in total can be transported over it. The goal is to maximize the total amount of transported goods.

Assume now that all capacities are integral. Is there always a maximum multi-commodity flow that is integral, i.e., such that each arc carries an integral amount of each commodity (so that we dont have to split containers)?