NOTALL3SAT = {p: $ is a 3CNF formula over n variables and has a satisf ying assignment in which every clause has at least one FALSE literal.}

Operations Research : Applications and Algorithms
4th Edition
ISBN:9780534380588
Author:Wayne L. Winston
Publisher:Wayne L. Winston
Chapter11: Nonlinear Programming
Section11.11: Separable Programming
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Problem 5. Define the language
{p: $ is a 3CNF formula over n variables and has a satisfying assignment
in which every clause has at least one FALSE literal.}
NOTALL3SAT =
In this problem you will prove that NOTALL3SAT is NP-Complete. If any truth assignment that satisfies p
has the property that at least one literal in every clause is FALSE, then we say that the truth assignment
is a NOTALL3-satisfying assignment.
A. Show that if all the variables in a NOTALL3-satisfying assignment are negated, then the result is
also a NOTALL3-satisfying assignment for p.
B. Suppose that o has m clauses. Now, transform $(x1, ... , xn) into another 3CNF formula
Y(x1, ... , Xn, yo, Y1, .. , Ym) as follows:
Convert the ith clause (z¡1 V zi2 V Zt3) of o into (z¡1 V Zi2 V Yi) ^ (ỹ, V zi3 V yo)
Note that p has 2m clauses, and that yo appears in m of them.
Show that if p has a NOTALL3-satisfying assignment, then p is satisfiable.
C. Show that if o is satisfiable then p has a NOTALL3-satisfying assignment.
D. Prove that NOTALL3SAT is NP-Complete.
Transcribed Image Text:Problem 5. Define the language {p: $ is a 3CNF formula over n variables and has a satisfying assignment in which every clause has at least one FALSE literal.} NOTALL3SAT = In this problem you will prove that NOTALL3SAT is NP-Complete. If any truth assignment that satisfies p has the property that at least one literal in every clause is FALSE, then we say that the truth assignment is a NOTALL3-satisfying assignment. A. Show that if all the variables in a NOTALL3-satisfying assignment are negated, then the result is also a NOTALL3-satisfying assignment for p. B. Suppose that o has m clauses. Now, transform $(x1, ... , xn) into another 3CNF formula Y(x1, ... , Xn, yo, Y1, .. , Ym) as follows: Convert the ith clause (z¡1 V zi2 V Zt3) of o into (z¡1 V Zi2 V Yi) ^ (ỹ, V zi3 V yo) Note that p has 2m clauses, and that yo appears in m of them. Show that if p has a NOTALL3-satisfying assignment, then p is satisfiable. C. Show that if o is satisfiable then p has a NOTALL3-satisfying assignment. D. Prove that NOTALL3SAT is NP-Complete.
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