Characterizing Multiterminal Flow Networks and Computing Flows in Networks of Bounded Treewidth1

We show that if a ow network has k input/output terminals (for the traditional maximum-ow problem, k = 2), its external ow pattern (the possible values of ow into and out of the terminals) has two characterizations of size independent of the total number of vertices: a set of 2 k + 1 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. For the case in which the underlying graph has bounded treewidth, we present sequential and parallel algorithms that can compute these characterizations as well as a ow consistent with any desired feasible external ow (including a maximum ow between two given terminals). For constant k, the sequential algorithm runs in O(n) time on 1 A preliminary version of this paper was presented at the 6th Annual ACM-SIAM Symposium on Discrete Algorithms in January 1995. 1 n-vertex networks, and the parallel algorithm runs in O(log n) time on an EREW PRAM with O(n=log n) processors if an explicit tree decomposition of the network is given; if not, known algorithms can compute a tree decomposition in O((log n) 2) time using O(n=(log n) 2) processors. The maximum-ow problem for general networks is P-complete and therefore unlikely to have fast and eecient parallel algorithms.