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Assuming P != NP

The euler diagram shows a part not part of P and NP-complete. I read on wikipedia that this set is called NP-Intermediate.

Euler Diagram

I have some doubts as to how are NPI problems defined?

Sam Radhakrishnan
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1 Answers1

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An NP-intermediate problem is a decision problem that

  • is in NP (that is, "yes" answers can be verified in polynomial time),
  • is not in P (that is, there is no polynomial-time algorithm for solving the problem), and
  • is not NP-complete.

That last criterion can be stated in a number of different ways. One way to say this is that there is no polynomial-time mapping reduction from SAT to that particular problem.

These problems are primarily of theoretical interest right now because we don't know if any NP-intermediate problems exist - if we could find one, we'd have a problem in NP that's not in P, meaning that P ≠ NP! However, they're interesting because if we can prove that P ≠ NP, then we know that there are some problems in NP that are too hard to be solved in polynomial time, but which aren't among the "hardest" of the hard problems in NP (the problems that are NP-complete).

In the event that P = NP, then there would not be any NP-intermediate problems because you couldn't have a problem in NP but not in P. If P ≠ NP, then Ladner's theorem guarantees at least one NP-intermediate problem exists, but does so by specifically constructing a problem that is highly artificial and designed solely to be NP-intermediate in that case. Right now, with a few exceptions (notably the graph isomorphism problem), all the problems we know of in NP are either squarely in P or known to be NP-complete.

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templatetypedef
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  • So a problem like factoring could be NP-Intermediate cause it is not P or NP-complete? – Sam Radhakrishnan Nov 24 '16 at 06:53
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    You'd have to be careful how you formulate it because that's not a decision problem, but something like "does n have a factor less than k?" might very well be NP-intermediate. – templatetypedef Nov 24 '16 at 08:11
  • [Integer factorization](https://en.wikipedia.org/wiki/Integer_factorization) would be the other superfamous unknown one. The wiki contains a list of a few more well known ones: https://en.wikipedia.org/wiki/NP-intermediate#List_of_problems_that_might_be_NP-intermediate . Also of interest: https://cstheory.stackexchange.com/questions/20930/why-are-so-few-natural-candidates-for-np-intermediate-status – Ciro Santilli OurBigBook.com Jul 08 '20 at 19:21
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    @CiroSantilli I was under the impression that integer factorization couldn’t be NP-intermediate because it isn’t a decision problem. Is there a way of naturally characterizing integer factorization in a way that makes it in NP? – templatetypedef Jul 08 '20 at 19:51
  • I always thought that what mattered for NP was: given this output (not necessarily yes or no), can you verify that it is correct, otherwise would be more limiting than feels needed. – Ciro Santilli OurBigBook.com Jul 08 '20 at 20:32
  • NP is the set of decision problems (yes/no) where, if the answer is yes, there exists a short certificate that proves the answer is yes. There is a related class FNP which are problems of the form “given x, find a y such that P(x, y) is true” given that P can be computed in time polynomial in the size of x. I think integer factoring is in FNP (y would either be the factors or a proof that x is prime), so perhaps integer factoring would be FNP-intermediate (?) – templatetypedef Jul 09 '20 at 00:20
  • After Googling, the wiki for integer seems to provide a decision version to which the search problem can be reduced in polynomial time: https://en.wikipedia.org/wiki/Integer_factorization "The decision problem version: given an integer N and an integer M with 1 < M < N, does N have a factor d with 1 < d ≤ M? This version is useful because most well studied complexity classes are defined as classes of decision problems, not function problems.", then bisect. Not sure if this counts as cheating. This method might work for every single problem? E.g. SAT as well. – Ciro Santilli OurBigBook.com Jul 09 '20 at 18:17
  • Some more research at: https://github.com/cirosantilli/cirosantilli.github.io/issues/63 but I'm not fully satisfied yet. – Ciro Santilli OurBigBook.com Jul 09 '20 at 18:17