Combinatorial Optimization - Homework I

K. Subramani LCSEE, West Virginia University, Morgantown, WV {ksmani@csee.wvu.edu}

1 Problems

1. Consider the feasibility version of the linear programming problem, in which you are given a matrix A_{m×n}, a vector b_{m×1} and asked, if there exists x ∈ ℜ^{n}, such that

A · x ≤ b.  (1)

In the integer programming counterpart, we are required to find an integral x, which satisfies System (1). Show that an oracle for the integer programming problem can be used to solve the linear programming problem in polynomial time.

2. Is the Fourier-Motzkin elimination procedure polynomially convergent? If yes, provide a proof; if no, provide a counterexample.

Solution:

□

3. In class, I showed that the same combinatorial optimization problem can have more than one integer programming formulation. Let F and G denote two formulations for a problem P. Formulation F is said to be stronger than formulation G, if the linear programming relaxation of F is a strict subset of the linear programming relaxation of G.

Consider the following two integer programming formulations of a problem:

F : 2 · x_{1} + 2 · x_{2} + x_{3} + x_{4} ≤ 2
x_{1} ≤ 1
x_{2} ≤ 1
x_{3} ≤ 1
x_{4} ≤ 1
x_{i} ≥ 0, i = 1, 2, 3, 4

G : x_{1} + x_{2} + x_{3} ≤ 1
x_{1} + x_{2} + x_{4} ≤ 1
x_{i} ≥ 0, i = 1, 2, 3, 4

Which one is stronger?

Solution: □

4. Let A denote an m × n matrix and b denote an m × 1 vector. Using the Strong Duality Theorem, prove that that either

∃x ∈ ℜ_{+}^{n} A · x ≤ b

or (mutually exclusively)

∃y ∈ ℜ_{+}^{m} y · A· ≤ 0
y · b < 0.

Solution: □

5. Let x_{1}, x_{2}, … x_{k} be points in ℜ^{n}. Argue that the following statements are equivalent:

(i) x_{1}, x_{2}, … x_{k} are affinely independent. (ii) (x_{2} − x_{1}), (x_{3} − x_{1}), … (x_{k} − x_{1}) are linearly independent. (iii) (x_{1}, 1), (x_{2}, 1), … (x_{k}, 1) are linearly independent. Solution: □