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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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