Homework 6

  • Assigned: Thursday, November 22, 2015
  • Due: Friday, December 11, 2015
  • General Instructions: 
    1. Please review the homework and grading policy outlined in the course information and homework page.
    2. Please use a cover sheet, as described in the homework page.
    3. You must also write down with whom you worked on the assignment, and any sources used.
    4. Start early, be concise but rigorous, and enjoy!
  • Specific Instructions:
    1. All logs are base 2 unless otherwise specified.
    2. Unless otherwise specified, all problems and exercises are from CLRS, Introduction to Algorithms, 3rd edition.
  • Problems:
    1. Exercises 25.2-6. 25.1-10. All pairs shortest paths.
    2. Exercises 26.1-2, 26.2-10. Maximum Flows. For 26.2-10, the problem is asking you to show that there always exsists a series of of at most E augmenting paths that leads to a maximum flow. You can assume that you already know the maximum flow, you have to explain how to construct the E paths.
    3. Problem 26-2. Minimum Path Cover. For this problem, be sure to explain how to use the solution to the max flow suggested in the hint to obtain a path cover and to prove that it is a minimum path cover.
    4. a) Show that the problem of determining whether two graphs are isomporphic is in NP. You can find the definition of graph isomorphism in the book. If you google graph isomorphism, you will find that there was recent progress on designing an algorithm for this problem. b) A hamiltonian path is a simple path that visits each vertex in a graph exactly once. Show that the Hamiltonian path problem, which takes a graph G and two vertices x and y and asks whether there is a hamiltonian path from x to y is in NP.
    5. Exercise 34.5-2. Integer linear programming.
    6. Problem 34-2. Bonnie and Clyde.