# Do any decision problems exist outside NP and NP-Hard?

This question asks about NP-hard problems that are not NP-complete. I'm wondering if there exist any decision problems that are neither NP nor NP-hard.

In order to be in NP, problems have to have a verifier that runs in polynomial time on a deterministic Turing machine. Obviously, all problems in P meet that criteria, but what about the problems with sub-exponential complexity? They do not belong to P and it's not obvious to me that they all have efficient deciders. And they certainly don't qualify for NP-complete.

I'm willing to believe that all decision problems are either NP or NP-hard or both, but nobody has actually said that (that I can find). I'm also willing to believe that such problems do exist, even if they are very contrived. Maybe someone more knowledgeable can put this issue to rest for me. Thanks.

Edit

I abused the term 'subexponential' in my question. In my mind it meant some problem with a complexity between exponential and polynomial like L-notation in this table. See the links in Raphael's answer for more details.

• Do you consider CO-NP to be a set of decision problems? – Aryabhata Apr 4 '13 at 5:06
• Our reference question about notions around NP may be a worthwhile read for you. – Raphael Apr 4 '13 at 6:29
• are you familiar with the time hierarchy theorem? it proves that problems of unlimited amount of time complexity exist. in other words the class "NP hard" also applies to "super hard" problems that provably require exponential time. – vzn Apr 4 '13 at 18:22
• I know that NP-hard is a huge category. I'm more interested in non-NP-hard problems. Specifically those that are non-NP. – Sebastian Goodman Apr 4 '13 at 18:53

NP contains all (decision) problems that are at most as hard as NP-complete problems with respect to Karp reductions.

NP-hard contains all (decision) problems that are at least as hard as NP-complete problems with respect to Karp reductions.

So, in the "Karp-reduction universe", NP and NP-hard do indeed cover everything. There may be orthogonal problem classes, though: for instance, co-NP-complete are neither in NP nor NP-hard (w.r.t. Karp reductions) if NP does not equal co-NP (which we don't know).

Aside from that, note that there are subexponential algorithms for NP-complete problems, but no known polynomial ones.

Thanks frafl for pointing out a critical flaw in the initial version of this answer. Readers of the old version may want to read this.

• I would say "cover everything comparable to NP" (in the partial order given by polytime Karp reductions; it's important to realize that this partial order is not a total order) – Sasho Nikolov Apr 4 '13 at 14:24
• So when you say w.r.t. Karp reductions, you mean problems where it is meaningful to talk about making Karp reductions on them to problems in NP? – Sebastian Goodman Apr 4 '13 at 18:55
• @SebastianGoodman: Under NP unequal co-NP, there is no Karp-reduction between NP-complete and co-NP-complete problems in either direction; the classes would be incomparable w.r.t. $\leq_p$, which is not a total order, as Sasho notes. – Raphael Apr 4 '13 at 23:28