Question

Suppose that you have a polynomial-time algorithm D for deciding whether a k-SAT formula Φ(x1, . . . , xn) on n variables is satisfiable. Design a polynomial-time algorithm that, when given a satisfiable such formula Φ as input, uses D to find a Boolean assignment a1, . . . , an ∈ {0, 1} that satisfies Φ (i.e., an assignment so that Φ(a1, . . . , an) = 1). Write pseudocode for your algorithm and explain why it is correct.

Answer 1

Boolean assignment that satisfies a satisfiable k-SAT formula, we can use a binary search approach with the algorithm D. The pseudocode for the algorithm is provided, along with an explanation of its correctness.

Step-by-step explanation:

Boolean assignment that satisfies a satisfiable k-SAT formula using the given polynomial-time algorithm D, we can use a binary search approach. We start with an initial assignment of all variables as 0. Then, we iteratively apply D to check if the formula is satisfiable with the current assignment. If D returns true, we know that the formula can be satisfied, and we keep the current assignment. If D returns false, we know that the formula is not satisfiable with the current assignment, and we change the assignment to all 1's. We repeat this process until we converge on an assignment that satisfies the formula.

The pseudocode for this algorithm:

def find_satisfying_assignment(Φ):

if D(Φ, assignment):

assignment = [1] * n

The correctness of this algorithm is based on the fact that if a k-SAT formula is satisfiable, there must exist a Boolean assignment that satisfies it. The binary search approach guarantees that we will eventually find such an assignment.