#### Solve the following by reducing to the original max-flow problem

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# Solve the following by reducing to the original max-flow problem. In other words, explain how to solve the new flow variant problem using an algorithm for solving max-flow as a black-box. Explain how to take an input for the new problem and define an input for the original max-flow problem. Then given a max-flow f ∗to this input you just defined, explain how to get the solution to the new problem. You should describe your design in words; no pseudocode. Justify the correctness of your design and state and justify the runtime. (a) (Vertex capacity) You are given a network {G = (V, E), ce, cv, s, t} where cv > 0, v ∈ V is de- fined as the capacity of each vertex and bounds the amount of flow that can enter the corresponding vertex. Your task is to find a max-flow that satisfies this new constraint, on top of the conservation of flow and the edge capacity constraints. (b) (Disjoint paths) Given a directed graph G = (V, E), and vertices s, t ∈ V , design an algorithm that outputs the maximum number of vertex-disjoint paths from s to t. Describe your design with words (no pseudocode), justify its correctness and state and analyse its runtime.

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