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For some time I've been trying to understand reduction of 3-SAT to MAX 2-SAT. I reviewed most of most popular books about computational complexity (Thomas Cormen, Papadimitriou) but I can't find an example, only how to do it theoreticaly and since I'm new to the topic I can't really make much progress. I don't fully understand it.

Thanks in advance for all of the comments trying to help.

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  • $\begingroup$ A simple proof can be found in this paper (Theorem 1.1 on page 240) It also shows an explicit construction of the MAX 2-SAT instance from the 3-SAT instance. $\endgroup$ – Reinhard Meier Feb 10 '19 at 14:40
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Assume you start with a 3-SAT instance with $m$ clauses. The usual reduction shows that each 3-CNF clause of a 3-SAT instance can be transformed into ten 2-CNF clauses such that a satisfying assignment to the 3-CNF clause can satisfy at most seven of the 2-CNF clauses. An unsatisfying assignment for the clause can satisfy at most only six of the produced 2-CNF clauses.

So after doing the transformation for all $m$ of the 3-CNF clauses the resulting MAX-2-SAT instance can have $7m$ of its clauses satisfied iff the original 3-CNF formula is satisfiable. Since determining the satisfiability of a 3-CNF formula is NP-hard, MAX-2-SAT must be NP-hard as well.

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