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We cannot deduce anything about the complexity of $g$. For a graph $G=(V,E)$, let $f_G$ be the function that maps each vertex set $S$ to $|S|$ if $S$ is an independent set and to $0$ otherwise. Obviously, $f_G$ is NP-hard to maximize. Now let $g_G$ be the function that maps each vertex set $S$ to $|V|$ if $S$ is nonempty and to $1$ if $S=\emptyset$. Then, ...


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If $F_3$ and $F_2$ are both monotone, satisfiability can be checked in polynomial time (or even in coNLOGTIME), as $F_3\land F_2$, which is also monotone, is satisfiable iff it is satisfied by the $\vec1$ assignment, that is, iff it does not contain the empty clause. If one of the formulas is allowed to be monotone and the other one negated monotone (i.e., ...


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You're right that the standard reduction from 3-SAT to 3D-matching (3DM) is not parsimonious. For the record, here's a sketch of a reduction that is parsimonious. It is obtained by composing parsimonious reductions from 3-SAT to 1-in-3-SAT, from 1-in-3-SAT to a problem we call 1+3DM, and from 1+3DM to 3DM. We sketch each of these next. Lemma 1. There is a ...


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