Algorithm Design

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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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