IEOR 6614, Spring 2016 : Homework 2

Assigned: Thursday, January 28, 2016
Due: Thursday, February 4, 2016

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 13.30. Most vital arc.
  2. Suppose, in the linear-time minimum spanning tree algorithm, we don't start with running 3 iterations the Baruvka algorithm, but instead start with 1. Is the resulting algorithm still linear time? Either prove that it is, or explain how the analysis from class breaks down. What if we run 2 iterations of the Baruvka algorithm? How about 4?
  3. Two integer programs for minimum spanning tree were presented in lecture. Show that the linear programming relaxation of the second integer program can have fractional extreme points, by constructing an example. Then show that every feasible solution to the first LP is also feasible for the second LP.
  4. Problem 4.42. Formulating shortest path problems.
  5. It is possible to reduce the running time of the shortest paths algorithm with radix heaps from O(m + n log (nC)) to O(m + n log C). Give the modified algorithm and analysis (Hint: one must show that fewer buckets are needed).

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    cliff@ieor.columbia.edu