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The theory of NP-completeness was initially built on Cook (polynomial-time Turing) reductions. Later, Karp introduced polynomial-time many-to-one reductions. A Cook reduction is more powerful than a Karp reduction since there is no restriction on the number of calls to the oracle. So, I am interested in NP-complete graph problem that does not have a known Karp reduction from a NP-complete problem.

Is there a natural graph problem known to be $NP$-complete only under Cook reduction, but not known to be NP-complete under Karp reductions?

Naturalness should disallow specific features of feasible solutions, for otherwise it is quite easy to start from well-known problem and make it a little easier by allowing specific features.

Yuval Filmus
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Mohammad Al-Turkistany
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  • "Known to be NP-compete only under Cook reductions" or "only known to be NP-complete under Cook reductions"? – David Richerby Dec 17 '13 at 21:05
  • @DavidRicherby The first one. – Mohammad Al-Turkistany Dec 17 '13 at 22:55
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    NON-HAMILTONICITY. – Yuval Filmus Dec 19 '13 at 05:38
  • NON-HAMILTONICITY is not known to be in $NP$. – Mohammad Al-Turkistany Dec 19 '13 at 05:58
  • According to Information System on Graph Classes and their Inclusions, Hamiltonian Path problem when restricted to 2-connected cubic planar graphs is not known to be $NP$-complete (I assume they mean under both types of reductions, Cook and Karp). Hamiltonian cycle problem is $NP$-complete on this graph class. – Mohammad Al-Turkistany Dec 19 '13 at 11:18
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    @MohammadAl-Turkistany HAMILTONICITY is NP-complete, so we don't expect that NON-HAMILTONICITY be NP-complete (unless NP=coNP). However, it is NP-hard with respect to Cook reductions. Now I see that this doesn't fit your description, because the problem is not in NP. – Yuval Filmus Dec 19 '13 at 16:06
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    See also this question: http://cstheory.stackexchange.com/questions/3333/contained-in-np-and-turing-reduction-from-an-np-complete-problem-rightarrow-n. – Yuval Filmus Dec 19 '13 at 16:38
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    its a known open problem, basically. they are conjectured to be different, see lutz/mayordomo ref/answer for this question many one vs turing reductions – vzn Dec 19 '13 at 17:39
  • This does not answer the question, but is somewhat related. If we lift your question to counting classes, then we know a "bit" more: counting number of perfect matchings in a bipartite graph is known to be #P-hard under polynomial time 1-Turing reductions, but it is open whether this holds for subtractive reductions. Moreover, it can not be #P-hard under parsimonious reductions unless P=NP; please see https://pdfs.semanticscholar.org/49bf/a5adc1e59ddff007abca1ee5a2ce536a80dc.pdf – Heyheyhey Sep 04 '17 at 20:30

1 Answers1

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I know an example of a problem that is missing two of the four features you ask for - it is not NP-complete, and it is not a problem on graphs.

Buchfuhrer and Umans (2011) show that the minimum equivalent expression problem in Boolean logic is complete for $\Sigma^P_2$ under polynomial-time Turing reductions.

Given a Boolean $(\wedge;\vee;\neg)$-formula $F$ and an integer $k$, is there an equivalent $(\wedge;\vee;\neg)$-formula of size at most $k$?

On p. 143, the authors state:

This provides a somewhat rare example of a natural problem for which a Turing reduction seems crucial (in the sense that we do not know of any simple modification or alternative methods that would give a many-one reduction).

They do not cite any other such examples from the literature. I suspect that "somewhat rare" is possibly an understatement.

References

David Buchfuhrer and Christopher Umans: "The complexity of Boolean formula minimization" Journal of Computer and System Sciences, Volume 77, Issue 1, January 2011, Pages 142-153

Hermann Gruber
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