CSOR 4231 07F: Homework 7

Required : 5 of the following 8 problems
Points: 10 pts per problem

General Instructions

  1. Please review the homework and grading policy outlined in the course information page.

  2. On the first page of your solution write-up, you must make explicit which problems are to be graded for "regular credit", which problems are to be graded for "extra credit", and which problems you did not attempt. Please use a table something like the following

    Problem01020304 0506070809...
    CreditRCRCRCECRC RCNARCRC...

    where "RC" is "regular credit", "EC" is "extra credit", and "NA" is "not attempted" (not applicable). Failure to do so will result in an arbitrary set of problems being graded for regular credit, no problems being graded for extra credit, and a five percent penalty assessment.

  3. You must also 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.
  4. Start early, be concise but rigorous, and enjoy!

Specific Instructions

  1. Unless otherwise specified, all problems and exercises are from CLRS, Introduction to Algorithms, 2nd edition.
  2. Please turn in each problem on a separate piece of paper(s) with your name on it.

Problems

  1. Let X be a problem that is in NP. Give an algorithm for problem X that runs in 2O(nk)) for some constant k.
  2. Suppose that someone gives you a polynomial time algorithm that, given a graph G and an integer k, tells you whether G has a clique of size k. Show how to use this algorithm to output the clique of size k in polynomial time.
  3. Exercise 34.5-2. Integer programming is NP-complete.
  4. Problem 34.1 Independent set.
  5. Problem 34.4. Scheduling
  6. Exercise 35.1-3. Vertex cover.
  7. Exercise 35.2-4. Bottleneck travelling salesman problem.
  8. Problem 35.5. Scheduling

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cliff@ieor.columbia.edu