**Assigned:**
Thursday, January 27, 2011

**Due:**
Thursday, February 3, 2011

## General Instructions

- Please review the
course information.
- You must write down with whom you worked on the assignment. If this
changes from problem to problem, then you should write down this
information separately with each problem.
- Numbered problems are all from the textbook
* Network Flows *.

## Problems

- Suppose we have a graph G and the edge weights are not
necessarily unique. Prove that a graph has a unique minimum spanning
tree if, for every cut of the graph, there is a unique minimum weight
edge crossing the cut. Prove that the converse is not true.
- Problem 13.30. Most vital arc.
- Suppose I have a graph G with n nodes and 5n edges. For a spanning tree F, we define h(F) to be the number of F-heavy edges in G. Give upper and lower
bounds on h(F) in terms of n. Repeat the problem for a complete graph (with n(n-1)/2 edges)
- Prove that the recurrence relation given in class,
T(n,m) = T(n/8,m/2) + T(n/8,m/4) + O(n+m) has solution O(n+m).

### Switch to:

cliff@ieor.columbia.edu