IEOR 6614, Spring 2007 : Homework

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 9.42. Converting a fractional flow to an integral flow.
2. Problem 10.6. Flow decomposition.
3. Suppose that given an n-node, m-edge graph G, you could find the most negative simple cycle in T(n,m) time. Analyze the algorithm where you push flow around the most negative cycle in the residual graph. You should state your time bounds in terms of T(n,m).
4. Although finding the most negative cycle is NP-hard, the following problem is solvable in polynomial time (via matching). Given a graph G, a cycle cover is a set of node disjoint cycles, such that each node is in at most one cycle. The minimum cycle cover is the cycle cover of minimum cost. Analyze the minimum cost flow algorithm in which you repeatedly find a minimum cycle cover in the residual graph and then push flow around each cycle in the cycle cover. You can assume that you can find a cycle cover in O(n³) time.

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• Homework 8
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