A T - Approach to Some Results on Cuts and Metrics

Let us commence by recalling the multicommodity flow problem and its 4 dual, the problem of packing metrics by cuts. A pair S s A, B of nonempty disjoint subsets of a finite set V is called a cut if B s V y A. Ž . Consider a network N s G, H, c, q consisting of a supply graph G s Ž . q 4 V, E endowed with a capacity function c: E a R j 0 , a demand graph Ž . q 4 H s X, F with X : V, and a demand function q: F a R j 0 . Denote 4 Ž . the edges of H by s t , . . . , s t . For a cut S s A, B of V let E S 1 1 m m denote the set of edges of G with one end in A and the other in B, and Ž . Ž . let c S s Ý c e be the capacity of the cut S. eg EŽS . The well-known multicommodity flow problem is to find flows f , . . . , f , 1 m where each f is a flow from s to t of value q , such that for each e g E i i i i Ž . the total flow through e does not exceed c e , or to establish that no such flows exist. By linear programming duality, a multicommodity flow exists if and only if