Consider the following problem. You are given a flow network with unit-capacity edges: It consists of a directed graph G = (V, E), a source s V, and a sink tV; and ce =1 for every e E. You are also given a parameter k.

The goal is to delete k edges so as to reduce the maximum s-t flow in G by as much as possible. In other words, you should find a set of edges F E so that |F| = k and the maximum s-t flow in G = (V, E F) is as small as possible subject to this.

Give a polynomial-time algorithm to solve this problem. Please give an algorithm with clear steps and give a FORMAL proof of the correctness of the algorithm and a FORMAL proof of the time efficiency of the algorithm.

2. Consider the following problem. You are given a flow network with unitcapacity edges: It consists of a directed graph G=(V,E), a source sV, and a sinktV; and ce=1 for every eE. You are also given a parameter k. The goal is to delete k edges so as to reduce the maximum s - t flow in G by as much as possible. In other words, you should find a set of edges FE so that F=k and the maximum st flow in G=(V,EF) is as small as possible subject to this. Give a polynomial-time algorithm to solve this