# IEOR 6614, Spring 2011 : Homework 2

Assigned: Thursday, January 27, 2011
Due: Thursday, February 3, 2011

## General Instructions

1. Please review the course information.
2. 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.
3. Numbered problems are all from the textbook Network Flows .

## Problems

1. 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.
2. Problem 13.30. Most vital arc.
3. 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)
4. 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