Question 6: 3-SAT with randomness
[8]
Let us recall the 3-SAT problem, in which we have a boolean formula \phi that is an AND of clauses
C1,..., Cm, where each Ci is the OR of three literals,¹ on the variables x1,..., xn.
(a) [4] Suppose we assign each xi to 0 or 1, uniformly at random, independent of the other xi. How
many clauses are satisfied in expectation?
(b) [4] Let the expected number in part (a) be m'. For a random assignment as above, give a lower
bound on the probability that the number of clauses satisfied is \ge m'. (To get credit, your bound
must be \ge 1/poly(m, n), as this means that trying poly(m, n) random assignments, we find an
assignment that satisfies \ge m' clauses, with high probability.)
