I am studying algorithms and there is a question in CLRS called the Half-SAT problem
We are given a 3-CNF formula with n variables and m clauses where m is even. We wish to determine whether there exists a truth assignment to the variables of f such that exactly half the clauses evaluate to zero and the other half to 1. Prove that the half 3-SAT problem is NP-Complete.
What is to stop me from taking any input f to 3-SAT and adding m clauses (y or y or y) where y is not in the original m clauses. The transformation of input and output is clearly polynomial time.
And, given a solution to 3-SAT, there must be a satisfying solution to Half-SAT (the solver sets y to false). If there is not a solution to 3-SAT this means that one of the original m clauses is not satisfied these clauses are disjoint from the new clause, so either m - 1 or 2m - 1 clauses are True.
Yet all the solutions online are more complicated which makes me think I am missing something obvious.