Please do all four problems. Show all work. No books or calculators allowed. You may use any result from class, the homeworks, or the texts, except where stated. You may use one sheet of handwritten notes. The exam lasts 110 minutes.

1. (20 points) For each of the following feasibility problems, indicate if it is polynomially solvable or -complete. (No partial credit. 4 points per part, -1 point for incorrect answers.)
   1. Node packing with lower bound: Given a graph G = (V,E) and an integer k, does there exist a node packing of cardinality at least k?

      Polynomially solvable -complete

   2. Perfect matching: Given a graph G = (V,E), does there exist a perfect matching?

      Polynomially solvable -complete

   3. Minimum spanning tree with upper bound Given a graph G = (V,E), edge weights w_e for e ∈ E, and an integer W, does there exist a spanning tree with total weight no larger than W?

      Polynomially solvable -complete

   4. Traveling salesman problem with upper bound: Given a complete graph on vertices V , with integer edge weights w_e, and a positive integer W, does there exist a traveling salesman tour of length no more than W?

      Polynomially solvable -complete

   5. Binary knapsack problem with lower bound: Given a ∈ ℤⁿ, c ∈ ℤⁿ, and scalars b and z, does there exist a binary x ∈ ℤⁿ with a^T x ≤ b and c^T x ≥ z?

      Polynomially solvable -complete

2. The binary variables x₁,…,x₅ satisfy the constraints
   1. (4 points) Show the constraints

      have Chvatal rank equal to one.

   2. (8 points) Show the valid constraint

      has Chvatal rank no greater than 3.

   3. (8 points) Show the valid constraint

      has Chvatal rank at least 2.

3. Consider the problem
   1. (10 points) Give the next level of the branch-and-bound tree using standard branch-and-bound. You need only give the optimal value at each node, together with the fathoming decision. Note that in an optimal solution to a relaxation, all the variables that take non binary values take the same value.

      

   2. (10 points) Give the tree you obtain if you use orbital branching. How many LP subproblems do you solve?
4. (20 points)
   
   1. (8 points) The nonnegative integer variable y₁ and the nonnegative continuous variable x must satisfy

      Give a valid linear constraint that is violated by the point x = 0, y = 4.

   2. (12 points) Let y₂ be a binary variable. Assume in addition that y₁ and y₂ must satisfy

      Lift the constraint you found in part 4a to give a valid constraint in x, y₁, and y₂.

5. The wheel W_n is a graph G = (V,E) with n + 1 vertices labelled 0, 1,…,n. It has 2n edges, of the form (1, 2), (2, 3), …, (n - 1,n), (1,n), and (0,i) for i = 1,…,n. Our feasible region S consists of all incidence vectors of Hamiltonian cycles on W_n.
   

   Every feasible solution satisfies the n + 1 degree constraints

   

   You may assume these equality constraints are linearly independent. (Hint: Let M be a square matrix with every entry equal to one. Let I be the identity matrix. You may assume that the columns of the matrix M - I are linearly independent.)

   1. (4 points) How many feasible solutions are there?
   2. (8 points) Show that the dimension of the feasible region is n - 1.
   3. (8 points) Show that the constraints x_e ≤ 1 define facets of conv(S) for the edges (1, 2), (2, 3), …, (n - 1,n), (1,n).