# IEOR 6614, Spring 2009 : Homework 6

Assigned: Thursday, February 26, 2009
Due: Thursday, March 5, 2009

## 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. Problem 7.4. An example of the push/relabel algorithm.
2. Problem 7.14. An example of the push/relabel algorithm.
3. Suppose that at some point in the execution of a push-relabel algorithm, there exists an integer 0 < k <= n-1 for which no vertex has d(v) = k. Show that all vertices with d(v) > k are on the source side of a minimum cut. If such a k exists, the gap heuristic updates every vertex v d(v) > k to set d(v) = max(d(v), n+1). Show that the resulting d(v) is a valid distance function, that is, d(v) <= d(w) + 1, if (v,w) is in the residual graph. (The gap heuristic is crucial in making implementations of the push-relabel method perform well in practice.)
4. Suppose that we have a flow network, but we know that each edge capacity is an integer of value at most k. Analyze runnning time of the generic push/relabel algorithm in terms of n, m and k.

### Switch to:

cliff@ieor.columbia.edu