CMPSCI 601: Theory of Computation

Offered through the PEEAS distance learning program

Homework Assignment #6

David Mix Barrington

Assignment posted Tue 12 August 2003

Due Mon 18 August 2003, by fax to 413-545-1249 ("attention Prof. Barrington") by 2:00 p.m. EDT

Make sure you have read the statement on academic honesty on the homework assignment index page.

Question 1 (20): Here is a revised version of the garbled problem 4 from HW#5. Let G be an undirected graph. A set of vertices C is a vertex cover for G if every edge in G is incident to at least one vertex in C. The MIN-VERTEX-COVER optimization problem is to input an undirected graph and output a vertex cover with as few vertices as possible.
The following algorithm is an approximation algorithm for MIN-VERTEX-COVER:
```
       Input: undirected graph G
       for all nodes v in G
          {mark(v) = false;}
       boolean done = false;
       while (!done)
          done = true;
          {for all edges (x,y) in G
              if ((mark(x) == false) && (mark(y) == false))
                 {mark(x) = true;
                  mark(y) = true;
                  done = false;
                  break;
                 }
          }
       return set of v such that mark(v) == true;
```
- (a,5) Explain why this algorithm must produce a vertex cover on any G and any order of the edges, and why it always runs in polynomial time.
- (b,5) Prove that this algorithm has an approximation ratio of at most 1/2 for the MIN-VERTEX-COVER approximation problem.
- (c,10) Find an example of a graph, and an order on the edges, for which this algorithm does no better than an approximation ratio of 1/2.
Question 2 (20): Let M be a poly-time nondeterministic Turing machine with a clock (so that it always halts in p(n) steps on input of length n). For any binary string w, let count_M(w) be the number of accepting paths of M on input w.
- (a,10) Describe a poly-space Turing machine that inputs w and outputs count_M(w).
- (b,10) Describe a poly-time alternating Turing machine A such that L(A) is the set {(w,t): t = count_M(w)}. (Typo corrected 15 August)
Question 3 (30): Let A be a language in BPP, that is, there exists a poly-time randomized Turing machine M such that for any w, the probability that M gets the right answer on w is at least 2/3. We want to show that for any number n, there is a poly-size circuit C_n such that for any w of length n, C_n correctly decides whether w is in A.
- (a,10) Explain how to construct a poly-time randomized Turing machine N such that for any w of size n, N decides whether w is in A with error probability at most 2^-(n+1). You may be vague about quantitative details, or quote them from a reference book, as long as the main point is clear.
- (b,10) Explain why there must exist a series S of coin flips for N such that for every w of size n, N with coin flips S correctly decides whether w is in A.
- (c,10) Explain how to use the result of (b) to build a poly-size circuit that correctly decides whether w is in A for any w of length n. Why is the family of such circuits not necessarily poly-time uniform?
- Question 4 (30): Consider the MAX-3-SAT optimization problem where the input is a 3-CNF formula and the goal is to find a setting of the variables that satisfies as many of the clauses as possible.
  - (a,5) Prove that if there is a poly-time algorithm solving MAX-3-SAT exactly, then P=NP.
  - (b,15) Prove the assertion in lecture that there is a poly-time algorithm that solves approximates MAX-3-SAT with threshold 1/2. Extra credit if you achieve approximation threshold 1/8 under the conditions for part (c) below.
  - (c,10) Prove that for any 3-CNF formula, there exists a setting of the variables that satisfies at least 7/8 of the clauses. (Here we assume that every clause has exactly three literals and that these literals are from distinct variables.)
- Question 5 (30 extra credit) (question added Thu 14 August) Recall that the language EQ_DFA is the set {(D,D'): D and D' are DFA's with the same input alphabet such that L(D) = L(D')}.
  
  (a,5) Prove that EQ_DFA is in the class NL.
  (b,10) Prove that EQ_DFA is not PSPACE-complete. Use no unproven assumptions. (Hint: Use the Space Hierarchy Theorem and Savitch's Theorem, which you may quote without proof.)
  (c,10) The language EQ_NFA = {(N,N'): N and N' are NFA's with the same input alphabet such that L(N) = L(N')}. Prove that EQ_NFA is in PSPACE. (In fact it is PSPACE-complete as it is P-equivalent to EQ_REX, see [P] exercise 20.2.13., but we won't prove this here.)
  (d,5) Explain how EQ_NFA can be PSPACE-complete and EQ_NFA not, when we know that NFA's can be converted to equivalent DFA's and vice versa.
Last modified 15 August 2003