Give a polynomial-time algorithm for the following minimization analogue of the Maximum-Flow Problem. You are given a directed graph G = (V, E), with a source s e V and sink t e V, and numbers (capacities) l(v, w) for each edge (v, w) e E. We define a flow f, and the value of a flow, as usual, requiring that all nodes except s and t satisfy flow conservation. However, the given numbers are lower bounds on edge flow—that is, they require that f (v, w) ≥ l(v, w) for every edge (v, w) e E, and there is no upper bound on flow values on edges.
(a) Give a polynomial-time algorithm that finds a feasible flow of minimum possible value.
(b) Prove an analogue of the Max-Flow Min-Cut Theorem for this problem (i.e., does min-flow = max-cut?).
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