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This is #P-hard, already for an arbitrary fixed constant approximation factor. As you noted, it allows you to approximate $|f(x)|$ for any GapP-function $f$, and therefore if $f$ is any #P (or GapP) function, it allows you to approximate $|f(x)-y|$ for a given $y$. With this, you can still compute $f(x)$ by a form of binary search. Specifically, fix a ...


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