PROBLEM 2 (30 points). In this exercise, we explore the effect of perturbations on the capacity of a single edge on the value of the maximum flow in a flow network with integral capacities. Consider the following types of edges in a flow network. An edge of a flow network is called critical if decreasing the capacity of this edge results in a decrease in the maximum flow. An edge of a flow network is called a bottleneck edge if increasing its capacity results in an increase in the maximum flow. Given a flow network G = - (V, E) with integer capacities ce > 0, prove or provide a counterexample for each of the following statements. (a) All critical edges are bottleneck edges. (b) A critical edge always exists. (c) A bottleneck edge always exists