Characterizations of k-Terminal Flow Networks and Computing Network Flows in Partial k-Trees

We show that if a ow network has k input/output terminals (for the traditional maximum-ow problem , k = 2), its external ow pattern (possible values of ow at the terminals) has two characterizations of size independent of the total number of vertices: a set of at most 2 k inequalities in k variables representing ow values at the terminals, and a mimicking network with at most 2 2 k vertices and the same external ow pattern as the original network. In the case that the underlying graph has bounded tree-width, we present a parallel algorithm that can compute these characterizations as well as an internal ow with any desired external ow pattern (including the maximum ow). The algorithm runs in O(log n) time on an EREW PRAM with O(n=log n) processors for constant k if an explicit decomposition of the graph is given; if not, the decomposition can be computed in NC. The maximum-ow problem for general networks is P-complete and therefore unlikely to be eeciently parallelizable.