On the NP-Hardness of Max-Not-2

We prove that, for any $\epsilon>0$, given a satisfiable instance of Max-NTW (Not-2), it is NP-hard to find an assignment that satisfies a fraction $\frac 58 +\epsilon$ of the constraints. This, up to the existence of $\epsilon$, matches the approximation ratio obtained by the trivial algorithm that just picks an assignment at random, and thus the result is tight. Said equivalently, the result proves that Max-NTW is approximation resistant on satisfiable instances, and this makes complete our understanding of arity three maximum constraint satisfaction problems with regards to approximation resistance.

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