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 in V, sink t in V, and nonnegative node capacities {c_v >= 0} for each v in V. Given a flow f in this graph, the flow through a node v is defined as f^(in)(v). We say that a flow is feasible if it satisfies the usual flow-conservation constraints and the node-capacity constraints: f^(in)(v) <= c_v 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.
We define the Escape Problem as follows. We are given a directed graph G=(V,E) (picture a network of roads). A certain collection of nodes x sub V are designated as populated nodes, and a certain other collection S sub V fe nodes. (Assume that x and S are disjoint.) In case the populated nodes Lenovo Exercises acity ated the sed ter ys sy ge nt s, h node-capacity constraints: f in v, 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 the populated nodes
