In a standard s-t Maximum-Flow Problem, we assume edges have capacities, and there is no limit on how much flow is allowed to pass through a node. In this problem, we consider the variant of the Maximum-Flow and Minimum-Cut problems with node capacities.
Let G = (V, E) be a directed graph, with source s e V, sink t e V, and nonnegative node capacities {cv ≥ 0} for each v ε V. Given a flow f in this graph, the flow though a node v is defined as fm(v). We say that a flow is feasible if it satisfies the usual flow-conservation constraints and the node-capacity constraints: fm(v) ≤ cv for all nodes.
Give a polynomial-time algorithm to find an s-t maximum flow in such a node-capacitated network. Define an s-t cut for node-capacitated networks, and show that the analogue of the Max-Flow Min-Cut Theorem holds true.
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