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I see the confusion, but I think the document you provided pretty well explains what is meant: solving MAXCUT on a graph $G$ is equivalent to finding the smallest value of $c$ such that $c-f_G(x)\geq 0$ for every $x\in \{-1,1\}^n$. As you write, it is trivially true that $c^*=\mathsf{OPT}(G)$ is the optimal value where this holds by definition, but for one, ...


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