Let ? be a 3cnf formula. A non-identical assignment, or nid-assignment, to the variables of ? is an assignment where each clause of ? contains two literals with different truth values, i.e., there cannot be a clause with the 3 true literals. Thus, a nid-assignment satisfies ? without assigning true to the 3 literals of any of the clauses.
(a) Show that the negation of a nid-assignment to ? is also a nid-assignment to ?.
(b) Let SATnid be the collection of 3cnf formulas that have a nid-assignment. Show that it is possible to obtain a polynomial-time reduction from 3SAT to SATnid by replacing each ci clause of the form (y1 ? y2 ? y3) by two clauses (y1 ? y2 ? zi) and (zi ? y3 ? b) where zi is a new variable for each ci clause, and b is a single additional new variable.
(c) Conclude that SATnid is NP-complete.
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