Algorithm Design

472
Chapter 8 NP and Computation!! Intractability
values at all nodes in K (as we did in the example of Figure 8.4). This set of
values clearly satisfies the SAT instance we constructed.
To argue the other direction, we suppose that the SAT instance we constmcted
is satisfiable. Consider a satisfying assignment for this instance, and
look at the values of the variables corresponding to the circuit K’s inputs. We
claim that these values constitute a satisfying assignment for the circuit K. To
see this, simply note that the SAT clauses ensure that the values assigned to
all nodes of K are the same as what the circuit computes for these nodes. In
particular, a value of 1 wil! be assigned to the output, and so the assignment
to inputs satisfies K.
Thus we have shown how to create a SAT instance that is equivalent to
the Circuit Satisfiability Problem. But we are not quite done, since our goal
was to create an instance of o-SK , which requires that all clauses have length
exactly B--in the instance we constructed, some clauses have lengths of 1 or 2.
So to finish the proof, we need to convert this instance of SAT to an equivalent
instance in which each clause has exactly three variables.
To do this, we create four new variables: zl, z2, z3, z4. The idea is to ensure
that in any satisfying assignment, we have z 1 = z2 = 0, and we do this by adding
the clauses (~ v z3 v z4), ~ V ~ V Z4), (~ V Z 3 V Z~ 4 , and (~ v ~ v ~ for each
of i = 1 and i = 2. Note that there is no way to satisfy all these claus6s unless
we set z 1 = z2 = 0.
Now consider a clause in the SAT instance we constructed that has a single
term t (where the term t can be either a variable or the negation of a variable).
We replace each such term by the clause (t v zl v z~). Similarly, we replace
each clause that has two terms, say, (t v t’), with the clause (t v t’ v z 0. The
resulting 3-SAT formula is clearly equivalent to the SAT formula with at most
three variables in each clause, and this finishes the proof. []
Using this NP-completeness result, and the sequence of reductions -
~ ~-SK T