In Section, we designed an approximation algorithm to within a factor of 7/8 for the MAX 3...

In Section, we designed an approximation algorithm to within a factor of 7/8 for the MAX 3-SAT Problem, where we assumed that each clause has terms associated with three different variables. In this problem, we will consider the analogous MAX SAT Problem: Given a set of clauses C1,…, Ck over a set of variables X = {x1,…, xn}, find a truth assignment satisfying as many of the clauses as possible. Each clause has at least one term in it, and all the variables in a single clause are distinct, but otherwise we do not make any assumptions on the length of the clauses: There may be clauses that have a lot of variables, and others may have just a single variable.

(a) First consider the randomized approximation algorithm we used for MAX 3-SAT, setting each variable independently to true or false with probability 1/2 each. Show that the expected number of clauses satisfied by this random assignment is at least k/2, that is, at least half of the clauses are satisfied in expectation. Give an example to show that there are MAX SAT instances such that no assignment satisfies more than half of the clauses.

(b) If we have a clause that consists only of a single term (e.g., a clause consisting just of x1, or just of ), then there is only a single way to satisfy it: We need to set the corresponding variable in the appropriate way. If we have two clauses such that one consists of just the term xi, and the other consists of just the negated term  then this is a pretty direct contradiction.

Assume that our instance has no such pair of "conflicting clauses"; that is, for no variable xi do we have both a clause  and a clause . Modify the randomized procedure above to improve the approximation factor from 1/2 to at least .6. That is, change the algorithm so that the expected number of clauses satisfied by the process is at least .6k.

(c) Give a randomized polynomial-time algorithm for the general MAX SAT Problem, so that the expected number of clauses satisfied by the algorithm is at least a .6 fraction of the maximum possible.

(Note that, by the example in part (a), there are instances where one cannot satisfy more than k/2 clauses; the point here is that we d still like an efficient algorithm that, in expectation, can satisfy a .6 fraction of the maximum that can be satisfied by an optimal assignment.)