Answered! (Flow decomposition 2-page limit – your solutions should fit on two sides of 1 page) A flow f is acyclic if there are no…

(Flow decomposition 2-page limit – your solutions should fit on two sides of 1 page) A flow f is acyclic if there are no directed cycles in the subgraph of edges with positive flow.

(a) Prove that every flow f has at least one corresponding acyclic flow that has the same value. (In other words, for every graph, at least one maximum flow is acyclic.)

(b) A path flow is a flow that gives positive values to a simple, directed path from source to sink. Prove that every acyclic flow is a finite combination of path flows.

(c) Some flows for a directed graph are not a combination of path flows. Give an example of one.