Homework 4

[Note: this webpage last modified Friday, 04-Feb-2011 19:44:51 EST]

This homework assignment will be handed out on Wednesday September 15 and is due by the beginning of class on the following Wednesday, so by 3pm Wednesday September 22. REMEMBER that your solutions should be explained in complete sentences such that your classmates will understand the solution (and can verify your proofs are correct) even if they have not solved the problems themselves. PLEASE TYPE YOUR solutions (in word, as text files, whatever you like), and send them to me by email. This way I will be able to read them regardless of your penmanship, and I do not have to carry them around.

  1. (15 Points) If L1 and L2 are problems in NP, which of L1 intersect L2, L1 union L2, and complement of L1 are in NP? For those that you claim are in NP, you must prove they are in NP.

    To get an intuition, consider some examples. For example let L1 be SAT and L2 be 3-Coloring. In this case, then x is a "yes" instance of L1 intersect L2 if x is a binary representation of both a satisfiable formula and a 3-colorable graph; x is in L1 union L2 if x is a binary representation of either a satisfiable formula or a 3-colorable graph; x is in the complement of L1 if x is a binary representation of a formula that has no satisfying assignment. For which of these can you give a poly-time verifier?

    For your answer/proof, you must let L1 and L2 be arbitrary NP problems, but you can consider examples to get intuition.

  2. Let NPNP be the set of decision problems that have proofs of membership that can be verified in polynomial time with access to an "oracle" for some decision problem in NP. That is, a decision problem L is in NPNP if there is a decision problem O in NP and a polynomial-time Turing machine V such that
    (i) for all x in L, there exists a proof of membership y such that VO(x, y) accepts, and
    (ii) for all x not in L, for any purported proof of membership y, VO(x, y) rejects.

    The notation VO means that the verifier V gets answers to questions of the form "is string z in O" for free.

    1. (10 Points) An input to the Graph Reconstruction problem is given by two collections of graphs (G1, G2, ..., Gk) and (H1, H2, ..., Hj). These collections are "yes" instances if there is some graph G such that every graph of the first set is isomorphic to some subgraph of G and every graph in the second set is not isomorphic to any subgraph of G. That is, (G1, G2, ..., Gk, H1, H2, ..., Hj) is a "yes" instance if there exists a graph G such that (i) for every 1 ≤ i ≤ k, there is a subgraph of G that is isomorphic to Gi, and (ii) for every 1 ≤ i ≤ j and for every subgraph G' of G, G' is not isomorphic to Hi.

      Show that the Graph Reconstruction problem is contained in NPNP. That is, show that there is a poly-time verifier / proof of membership if we allow the verifier to get answers to NP problems for free.

      To start with, what should the witness/proof/certificate be?

    2. (10 Points) Show that if P = NP, then NPNP = P.

      In particular, if P=NP, then Graph Reconstruction is contained in P. As a warmup you can think about why P=NP implies this problem is in P.

    Note, for your information only - this will not help you in answering the questions. The process for defining NPNP can be further generalized to define classes such as NPNPNP, etc. It can be shown that for any constant number of "towers", this class collapses to P if P = NP. The important point here is that if P = NP, then we would have polynomial-time algorithms for all NP problems, but we would also have polynomial-time algorithms for all of these other problems as well (which seem even harder to solve than generic NP problems).

  3. (20 Points) Choose any NP-complete or PSPACE-complete problem that we have not covered in class. You will present to the class that this problem is NP-complete or PSPACE-complete on Wednesday when the homework is due, and you will have approximately 5 minutes to do so. You may work in groups of two or three. You must get approval from me on the problem that you will present. You will provide a citation to the class about where you got either the proof or proof idea for the completeness proof. You will also include a brief "main idea" type explanation of the NP-completeness reduction in your homework 4 submission.