Half-integral five-terminus flows

Suppose that G is an undirected graph whose edges have nonnegative integral capacities, that st, . . . ,s~ are k distinguished vertices in G, and that dti, I si<jl k, are nonnegative integral demands. We consider the problem of existence of an admissible multicommodity flow P= {Fii: 1 si<j= k) in which each Fii is a flow of value dii between si and sj. It is known that if kr5, then satisfying the obvious requirements of Ford-Fulkerson’s type (concerning cuts of G) does not guarantee, in general, that such a multicommodity flow exists. We give a combinatorial criterion of solvability of this problem for k= 5 and prove that if k=S and the problem has a solution, then it has also a half-integral solution. This generalizes a number of known results on half-integral multicommodity flows. Also other results on multicommodity flows and metrics are . presented.