On a Pooling Problem with Fixed Network Size

The computational challenge offered by most traditional network flow models is modest, and large scale instances can be solved fast. The challenge becomes more serious if the composition of the flow has to be taken into account. This is in particular true for the pooling problem, where the relative content of certain flow components is restricted. Flow entering the network at the source nodes has a given composition, whereas the composition in other nodes is determined by the composition of entering flows. At the network terminals, the flow composition is subject to restrictions. The pooling problem is strongly NP-hard. It was recently shown that at least weak NP-hardness persists in an instance class with only two sources and terminals, and one flow component subject to restrictions. Such instances were also shown to be solvable in pseudo-polynomial time if a particular fixed-size instance class, called atomic instances, can be solved in terms of a constant number of arithmetic operations. This work proves existence of such a closed-form solution to atomic instances.