For the graph below write the math programming models for the specified problems. You don't have to write complete models, but show enough to illustrate your approach.

a.   Write a math-programming model that would find the maximum cardinality matching.

b.   Write a math-programming model that would find the minimum cardinality cover.

c.   Say edge k has the length c(_k). Write a math programming model that would solve the traveling salesman problem.

d.   Say edge k has the length c(_k). Write a math programming model that would find the shortest path from node 1 to node 9. The path does not have to go through every node. Use a variable x_(k) = 1 if edge k is included in the path and 0 otherwise.