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Let $F$ be a CNF formula. Let $l$ be one of $F$'s literals.

Question

Which is the complexity of determining whether $l$ is a backbone literal or not? The obvious way to do that is to propagate $\lnot l$ on $F$, obtaining $F'$: $l$ is then a backbone literal of $F$ if and only if $F'$ is unsatisfiable. I'm asking if there is another way.

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  • $\begingroup$ Isn't that just SAT of $F\land\lnot l$? That is, it's NP-complete. Am I missing something? $\endgroup$ Apr 1 '11 at 17:04
  • $\begingroup$ @Radu GRIGore: What you say is of course right, it is the obvious and natural way to do that: just reduce the question to SAT (note this doesn't mean the question is NP-complete too). I wonder if there is another way. $\endgroup$ Apr 1 '11 at 17:10
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    $\begingroup$ Backbone literals look very similar to frozen variables. There are ways to determine/approximate them in random k-CNF literature. $\endgroup$
    – Kaveh
    Apr 1 '11 at 21:09
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    $\begingroup$ For practical algorithms, I believe Marques-Silva, Janota, and Lynce is a good place to look. $\endgroup$ Apr 1 '11 at 22:45
  • $\begingroup$ @Radu: Thanks for the pointer to that paper. I'm reading it right now. $\endgroup$ Apr 2 '11 at 9:11
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The complexity is co-NP complete because you're converting to unsatisfiability of $F \land \lnot l$. I have a proof of the completeness in my PhD thesis, Claim 2 (if I may blatantly advertise myself).

There's a small problem for unsatisfiable formulas, if your definition of a backbone means that all the literals are backbone for an unsatisfiable formula, then the problem stays in co-NP. If, however, the set of backbone literals is empty for an unsatisfiable formula, you also need NP.

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  • $\begingroup$ I prefer the definition where all the literals are backbone for an unsatisfiable formula. $\endgroup$ Apr 18 '11 at 13:00

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