In a standard minimum s-t cut problem, we assume that all capacities are nonnegative; allowing an arbitrary set of positive and negative capacities results in a problem that is computationally much more difficult. However, as we’ll see here, it is possible to relax the nonnegativity requirement a little and still have a problem that can be solved in polynomial time. Let G = (V, E) be a directed graph, with source s ∈ V , sink t ∈ V , and edge capacities {ce}. Suppose that for every edge e that has neither s nor t as an endpoint, we have ce ≥ 0. Thus ce can be negative for edges that have at least one end equal to either s or t. Give a polynomial-time algorithm to find an s-t cut of minimum value in such a graph. (Despite the new nonnegativity requirements, we still define the value of an s-t cut (A, B):

X (u,v)∈E u∈A, v∈B c(u,v)

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