Name: |

MATP6620/ISYE6770

Integer and Combinatorial Optimization

Spring 2019

Midterm Exam, Friday, March 29, 2019.

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.

Q1 | /20 | |

Q2 | /20 | |

Q3 | /20 | |

Q4 | /20 | |

Q5 | /20 | |

Total | /100 |

- (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.)
- 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

- Perfect matching: Given a graph G = (V,E), does there exist a perfect matching?
Polynomially solvable -complete

- 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

- 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

- Binary knapsack problem with lower bound: Given a ∈ ℤ
^{n}, c ∈ ℤ^{n}, and scalars b and z, does there exist a binary x ∈ ℤ^{n}with a^{T }x ≤ b and c^{T }x ≥ z?Polynomially solvable -complete

- 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?
- The binary variables x
_{1},…,x_{5}satisfy the constraints- (4 points) Show the constraints
have Chvatal rank equal to one.

- (8 points) Show the valid constraint
has Chvatal rank no greater than 3.

- (8 points) Show the valid constraint
has Chvatal rank at least 2.

- (4 points) Show the constraints
- Consider the problem
- (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.
- (10 points) Give the tree you obtain if you use orbital branching. How many LP subproblems do you solve?

- (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.
- (20 points)
- (8 points) The nonnegative integer variable y
_{1}and the nonnegative continuous variable x must satisfyGive a valid linear constraint that is violated by the point x = 0, y = 4.

- (12 points) Let y
_{2}be a binary variable. Assume in addition that y_{1}and y_{2}must satisfyLift the constraint you found in part 4a to give a valid constraint in x, y

_{1}, and y_{2}.

- (8 points) The nonnegative integer variable y
- 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.)

- (4 points) How many feasible solutions are there?
- (8 points) Show that the dimension of the feasible region is n - 1.
- (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).