where f is the max-flow of the UMFP. (Henceforth, we will refer to the quantity C(U, [bar]U)/|U||[bar]U| as the ratio cost of a cut <U, [bar]U>.) Since the min-cut of a UMFP is
where f is the max-flow and S is the min-cut of the UMFP.
The algorithm for finding a cut with small ratio cost is based on the linear programming dual of the UMFP. In general, the dual of a multicommodity flow problem for a graph G is the problem of apportioning a fixed amount of weight (where weights are thought of as distances) to the edges of G so as to maximize the cumulative distance between the source/sink pairs.
From the duality theory of linear programming, we know that an optimal distance function results in a total weight that is equal to the max-flow of the UMFP. Hence, by solving the dual UMFP, we can find distances d(e) that satisfy Eq.
The fact that our algorithm uses the dual UMFP to find a small cut should not be completely surprising, given the well-known relationship between the min-cut and the dual of a single commodity flow problem.
The demand for the commodity between nodes u and v is then set to be [Pi](u)[Pi](v).(7) The UMFP is a special case of a PMFP for which [Pi](u) = 1 for all nodes u [element of] V.
In a directed UMFP, we will assume that the demand from u to v is 1 for each u [is not equal to] v.
The max-flow and min-cut of a directed UMFP are related by the following theorem.
where f is the max-flow and S is the rain-cut of the UMFP.
As a consequence of this work, we have found that the max-flow of a UMFP or PMFP is nearly as large as the limit implied by the min-cut.
The dual of the short-path UMFP is the same as before except that the distance between two nodes in the distance constraint is computed using only paths with at most L edges in G.
 give an alternative approach to finding balanced separators using a generalized version of the dual to the UMFP. In addition, M.