Combinatorial Optimization

K. Subramani, LCSEE, West Virginia University

March 20, 2013

1. Does greedy algorithm work for more general independence systems than matroids? Greedy characterization theorem.

2. Problem 1 - Nathan.

3. Problem 2 -

4. Problem 3 -

5. Problem 4 - Let A denote the TU matrix. The following facts of determinants must be known:

(a) Elementary row operations do not change the magnitude of the determinant.

Let B be a square submatrix of A. Let B′ denote B after pivoting. Let a_{ij} denote the pivot element. Three cases:

(a) B contains some part of i^{th} row.

(b) B contains no part of i^{h} row, but some part of j^{th} column.

(c) B does not contain any part of the i^{th} row or the j^{th} column.

6. Problem 5 - First observe that,

Lemma 0.1 Let A · x ≤ b denote a TDI-system, where A is rational and b is integral. Then the polyhedron P ≡ x : A · x ≤ b is integral.

Proof: A polyhedron is said to be integral, if all its vertices (extreme points) are integral. All that we need to show is that max{c·x : x ∈ P} is attained by an integral vector for each c, for which the maximum is finite. Indeed, we can restrict ourselves to integral c! Why?

Consider the system:

z = max{c · x : A · x ≤ b}

and its dual

w = min{y · b : y · A = c, y ≥ 0}

Assume that z is finite. We need to show that x is integral.

In our case, we are given that A · x ≤ b is TDI, and that b and c are both integral.

By strong duality theorem w = z. Since A · x ≤ b is TDI, and c is integral, we must have y is integral at optimality, i.e., y^{∗} is integral.

Since y^{∗} is integral and b is integral, so is w. It follows that z is integral. □