IEOR 6614, Spring 2012 : Homework 3

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. In class, we described a linear program for MST that had a variable x_e which was 1 if edge e is in the MST, a constraint that the tree has n-1 edges, and a constraint for each subset of the vertices, saying that a subset S of size |S| has at most |S|-1 tree edges. Write down the dual of this linear program. Then write down the complimentary slackness conditions. Then do the same for a linear program where the constraints are a) n-1 edges b) for every cut in the graph, there is at least one edge crossing the cut.
2. Given a graph G=(V,E) and a subset of edges E', give a linear-time algorithm that returns a graph in which all the edges of E' are contracted.
3. Given a directed acyclic graph G, with source c let ct(v) be the number of distinct paths from s to v. Give an efficient algorithm to compute ct(v) for all vertices v.
4. Problem 5.18. Reoptimizing shortest paths.
5. Problem 4.6. Cluster analysis.

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• Homework 2
• Homework 4