Recall the MAX

$-$

$3$

$-$

SAT problem: given a set of clauses

$$

$1$

$,$

$$

$2$

$,$

dots,

$$

$$

$,$

$$each a disjunction of

$3$

$$terms over a set of variables

$$

$=$

$\{$

$$

$1$

$,$

$$

$2$

$,$

dots,

$$

$$

$\}$

$,$

$$find a truth assignment of the variables that satisfies as many of the

$$

$$clauses as possible.

$($

a

$)$

$($

$3$

$$points

$)$

$$In class we saw a very simple, polynomial

$-$

time, randomized algorithm for achieving a

$7$

$8$

$$approximation

$($

in expectation

$)$

$$to the optimal solution of MAX

$-$

$3$

$-$

SAT. What was the algorithm? Give either a description of it in English or provide some brief pseudocode for it

$.$

$($

b

$)$

$($

$2$

$$points

$)$

$$We noted that an immediate consequence of this result in part

$($

a

$)$

$$is the following fact: For every instance of

$3$

$-$

SAT, there exists an assignment of the boolean variables that satisfies at least a

$7$

$8$

$$fraction of all the clauses. That is

$,$

$$if

$$

$$is the number of clauses, there is an assignment that satisfies at least

$7$

$$

$8$

$$of them.