Is it possible to reduce MaxUNSAT to MaxSAT in a polynomial way ?

When considering the MaxSAT problem, one often considers also the MinUNSAT problem, which is almost the same. And for a propositional formula f in CNF it holds:

|f| = MaxSAT(f) + MinUNSAT(f)

where |f| is the number of clauses of f.

When considering MaxUNSAT and the corresponding MinSAT problem, the same relationship holds:

|f| = MaxUNSAT(f) + MinSAT(f)

Now, I was wondering if there is also a relationship between those two pairs, e.g. to reduce MaxSAT to MaxUNSAT or MinSAT (or the other way round) ?

Unfortunately, I could not figure out one by myself. And maybe there is none ?

Update 1: Inspired by Yuval Filmus's answer, I will give a reduction for my question.

Reduction from MaxUNSAT to its corresponding decision problem:

Let $\phi = {C_1, ..., C_m}$ a set of clauses over the variables $x_1, ..., x_n$, then it holds: $$MaxUNSAT(\phi) = BinarySearch(0, |\phi|, MaxUNSAT(\phi, k) )$$ with

BinarySearch(start, end, CompareProcedure( )):

Searches for the element e between start and end so that CompareProcedure(e)=true and CompareProcedures(e+1)=false


$$MaxUNSAT(\phi, k) := \exists v\in\{0, 1\}^n:\sum_{i=1}^m 1 - I_v(C_i) \geq k$$ where $I_v$ is the interpretation of a propositional formula under assignment $v$.

Reduction from decision problem $MaxUNSAT(\phi, k)$ to SAT:

One can reduce the devision problem $MaxUNSAT(\phi, k)$ to the SAT problem by adding blocking variables to each clause and adding a cardinality constraint as propositional formula to limit the number of used clauses with help of the blocking variables.

I can describe this in more detail, if needed.


One can reduce the MaxUNSAT problem to the SAT problem and then solve the SAT problem with the MaxSAT problem. This is a reduction that works in polynomial time.

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    $\begingroup$ I do not think that it is fair to change the question significantly after you receive an answer. $\endgroup$ Sep 24, 2012 at 16:59
  • $\begingroup$ I do not think it has changed significantly. The question just changed from "How to reduce ... " to "How to reduce it in a more direct way". But if you want, I can break this question into two questions (open up a new question for the second one) to make things more clear/fair ? $\endgroup$ Sep 24, 2012 at 21:27
  • $\begingroup$ I personally think so, because Yuval has already answered your original question and it seems that the purpose of your edit in revision 5 is to make his answer invalid. Other people may think otherwise, though. $\endgroup$ Sep 24, 2012 at 21:44
  • $\begingroup$ Yuval Filmus did not answer my question. He proved that there exists a reduction, but my question was not "Is it possible to reduce" but "How to reduce". So either way, his answer helped me to find a reduction, but did not provide a solution by itself. Therefore, my change did not touch his answer. But I understand the confusion of the change and to make things clear/fair again, I will open up a new question and rephrase this one to its original. $\endgroup$ Sep 24, 2012 at 21:54

1 Answer 1


Both MaxSAT and MaxUNSAT are NP-complete, and so can be reduced to one another. MaxSAT is NP-hard by (trivial) reduction from SAT. To see that MaxUNSAT is NP-hard, we follow Creignou [1] and reduce from MaxCUT. Given an instance $G=(V,E)$ of MaxCUT, consider the instance $$ \bigwedge_{(x,y) \in E} (x \lor \lnot y) \land (\lnot x \lor y). $$ Each cut corresponds to a $0/1$ assignment to the vertices. If an edge $(x,y)$ is separated by a cut, then one of the two corresponding clauses will be unsatisfied, and otherwise none will be unsatisfied.

[1] Creignou, A dichotomy theorem for maximum generalized satisfiability problems, JCSS 51, 511-522 (1995).

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    $\begingroup$ Thank you. I updated my question with help of your answer. But you have to be careful with the term NP-complete when talking about optimization problems (which are not decision problems). $\endgroup$ Sep 24, 2012 at 8:23

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