Combinatorial Optimization - Homework II

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

1 Instructions

1. The homework is due on March 7, in class.

2. Each question is worth 4 points.

3. Attempt as many problems as you can. You will be given partial credit, as per the policy discussed in class.

4. Some of the problems require at least a passing familiarity with the Tableau method for implementing the simplex algorithm.

5. The work must be entirely your own. You are expressly prohibited from consulting with colleagues or the internet (with the exception of the material on the course website).

2 Problems

1. Let S denote a convex set and let x denote a point in S. Argue that x is an extreme point of S, if and only if S − {x} is convex.

2. Use the Tableau method to prove the Strong Duality theorem.

3. Use the Tableau method to prove the Strong Basis theorem.

4. Prove that if A is a (Totally Unimodular) TU matrix, then any matrix obtained from A, by performing a Gauss-Jordan pivot is also TU.

5. A polyhedral system A · x ≤ b is said to be Totally Dual Integral (TDI), if the minimum in the Linear Programming

duality equation

max{c · x : A · x ≤ b} = min{y · b : y · A = c, y ≥ 0}

has an integral solution y for each integral c, for which the minimum is finite.

Consider the following conjecture:

Conjecture 2.1 An integral matrix A is Totally Unimodular, if and only if the system A · x ≤ b, x ≥ 0 is TDI, for each vector b. Is the conjecture valid?