CMPSCI 601: Theory of Computation

Offered through the PEEAS distance learning program

Homework Assignment #7

David Mix Barrington

Assignment posted Thu 11 Dec 2003

Due Wed 17 Dec 2003, 2:00 p.m. by fax to 413-545-1249, attention Prof. Barrington.

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

This assignment is about half the size of the first five, since you have so little time for it.

This assigment is also optional in that it will be counted for your final grade if and only if this is to your advantage.

• Question 1 (20): In Lecture 25 the class ZPP is defined to be the intersection of RP and co-RP. Define the class LVP (Las Vegas poly time) to be the set of problems decidable by randomized algorithms that always get the right answer and run in expected polynomial time. Prove the assertion in lecture 25 that ZPP equals LVP.

• Question 2 (15): In Lecture 26 the class PCP[log n, O(1)] is defined and the PCP Theorem, that this class equals NP, is quoted. Prove the easy half of the PCP Theorem, that PCP[log n, O(1)] is contained in NP.

• Question 3 (15): Explain in your own words (perhaps with the help of a reference) a greedy deterministic algorithm that approximates MAX-3-SAT with approximation ratio 1/7. (That is, the algorithm should satisfy at least 7/8 the number of clauses that are possible to satisfy.) Here it is important that the clauses each contain three distinct variable.