Formulate the following as optimization problems, giving in each case a description of the set of feasible solutions F and the objective function c. Create a small instance for each problem, and state F and c.

(a) Find the longest path in between any two nodes in a graph.

(b) Find a cylinder with a given surface area A that has the largest volume.

(c) Find a path on a graph that visits every node of the graph exactly once.

(d) Given an n n chessboard, find the maximum number of rooks that can be placed such that none of them can attack to each other.