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.

   Solution:

   1. We take 0.5 of the original constraint and round down:

   2. We use the rank 1 inequality from part (a) to construct the rank 2 inequalities with sums of 4 variables, and then combine these to give the desired inequality, so it has rank no greater than 3:

      For any 1 ≤ i < j < k < l ≤ 5, we take of the four inequaities from part (a) involving these variables:

      

      We now take of these 5 inequalities to give

      

      as required.

   3. The optimal solution to the LP relaxation with objective function max ∑ _j=1⁵x _j is x = (,,,,), with value 4. Since this value is greater than 3, the desired inequality cannot have Chvatal rank 1.
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?
      

      Thus, we need only solve 7 linear programs.

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.

   Solution:

   1. The mixed integer cut is

      where f = , the fractional part of the right hand side. Can rewrite this as

      

   2. The lifting subproblem is

      Optimal solution is x = 3, y₁ = 1, y₂ = 1, with value 11. Taking the difference between this value and the right hand side of 5 for the constraint shows that y₂ has coefficient -6, so the lifted constraint is

      

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).

   Solution:

   1. There are n feasible solutions: omit one of the rim edges.
   2. If we order the n rim edges first, the first n components of the incidence vectors of our n solutions give the columns of the matrix M - I. Thus, these incidence vectors are linearly independent, so they are affinely independent. Therefore, the dimension of the feasible region is at least n - 1.

      The region S satisfies n + 1 linearly independent equality constraint, so the dimension of the feasible region is no greater than 2n - (n + 1) = n - 1.

   3. Just one of the feasible solutions does not satisfy x_e = 1 for the listed edges. Hence, we have n-1 feasible solutions that satisfy the constraint at equality. From part (b), these feasible solutions are linearly independent, so they are affinely independent. Thus, the dimension of the face is at least n - 2, so it is a facet.