IEOR 6614, Spring 2016 : Homework 5

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. Write down the maximum flow problem as a linear program. Take the dual. Explain how the dual can be interpreted as a minimum cut problem.
2. Problem 7.12. Critical arcs.
3. Problem 7.20. (The highest-label algorithm applies discharge to the active vertex with the highest distance label.)
4. Problem 7.24. Variants of excess scaling.
5. 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 for which d(v) > k to set d(v) to max(d(v), n+1). Show that the d is still a valid distance function. (The gap heuristic is crucial in making implementations of the push-relabel method perform well in practice.)

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