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Is there a natural Karp reduction from Independent Set to SAT ? That is, a reduction that does not rely on the Turing machine (as the case in proof of Cook's theorem) but the combinatorial structure.

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Yes, just think how you would use a SAT solver to solve the independent set problem.

If you have one variable per node, it should be clear how to write constraints that guarantee that variables that are true indeed form an independent set. I think the only non-trivial part is counting: how to make sure that your independent set is sufficiently large. For that you can use BDD-based techniques; see e.g. Section 5.3 of this paper.

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  • $\begingroup$ Any chance you could update the broken link / add citation info? I can't access it and can't tell what it's supposed to be pointing to. Thanks in advance! $\endgroup$ – Joshua Grochow Sep 15 at 19:42
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    $\begingroup$ @JoshuaGrochow Fixed the link! $\endgroup$ – Jukka Suomela Sep 15 at 20:39
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    $\begingroup$ For some of these problems it is easier to reduce to Circuit SAT since one is presumably used to the idea of implementing arithmetic naturally via circuits and gates. And then use a Karp reduction from Circuit SAT to SAT. This is assuming that one wants to "see" a somewhat natural reduction rather than use a SAT solver etc. $\endgroup$ – Chandra Chekuri Sep 16 at 0:56

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