"For each of the following two statements, decide whether it is true or false. If true, give a short explanation. If false provide a counterexample.
a) Suppose we are given an instance of the Minimum Spanning Tree Problem on a graph G, with edge costs that are all positive and distinct. Let T be a minimum spanning tree for this instance. Now, suppose we replace each edge cost Ce by its square, Ce^2, thereby creating a new instance of the problem with the same graph but different costs.
True or false? T must still be a minimum spanning tree for this new instance.
b) Suppose we are given an instance of the Shortest s-t Path problem on a directed graph G. We assume that all edge costs are positive and distinct. Let P be a minimum cost s-t path for this instance. Now suppose we replace each edge cost Ce with its square, Ce^2, thereby creating a new instance of the problem with the same graph but different costs.
True or false? P must still be a minimum cost s-t path for this new instance."