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Questions tagged [np]

Questions about decision problems that can be solved on nondeterministic Turing machines in time polynomial in the length of the input.

NP is the class of decision problems that can be solved on nondeterministic Turing machines in time polynomial in the length of the input. Equivalently, these are decision problems where every "yes" instance has a proof of this fact, which has length polynomial in the size of the instance and can be verified deterministically in polynomial time.

One of the major open problems in computer science is whether P=NP. That is, whether deterministic polynomial time Turing machines have the same power as nondeterministic ones. Or, informally, whether being able to easily compute the answer to a problem (P) is the same as being able to easily verify an answer, in the sense given above. The Clay Mathematics Institute has offered a million-dollar prize for the resolution of this question.

For more information, see our reference question.

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A "natural" decidable problem not in $\mathsf{NP}$?

Are there any "natural" examples of decidable problems that are definitively known not to be in NP? The decidable languages I know of that are not contained in NP are usually derived from the time hierarchy theorem, which produces "artificial"…
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NP-hard - Definition

The definition of NP-hard problem in Wikipedia is : NP-hardness (non-deterministic polynomial-time hard), in computational complexity theory, is a class of problems that are, informally, "at least as hard as the hardest problems in NP". There are…
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Is knowing ALL factors of a large number an NP?

Say I have a number 45 What are the factors of 45? 1, 3, 5, 9, 15, 45 Is there any other factors? I know that if I want to find some non trivial factor of some number, say 45, I know it's in NP. I can non deterministically choose some 2 numbers and…
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Is everything that is turing computable an element of NPSPACE?

So, if there is something that is turing computable, I know it is in NP (is that right?). If M is computable, does it mean M is in NP? M computable IFF in NP (is that true)? And does all computable machines exist under NPSPACE?
Chung Chen
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Confusion about proving CIRCUIT-SAT is in NP

My textbook states that CIRCUIT-SAT is in NP because you can guess an input and verify it in polynomial time. My textbook also states that a problem is in NP if and only it has a polynomial-time verifier V(I, X) such that: if I is a YES instance,…
user81356
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Why IS is not in NL, only in NP?

All we need to do is guess $k$ vertices. We look at vertex $v_1$, and make sure $v_1$ is not connected to $v_2...v_k$. Then, we "throw" $v_1$, and look at $v_2$. We do this to all vertices. Meaning that we only need to guess $k$ vertices, and in our…
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Are there any known "hard" instances for NP-Complete Problems

Are there any known "hard" instances for NP-Complete Problems, or are there no general hard instances. So for different algorithms different instances are hard?
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Is it true that all languages which have polynomial circuits are in PSPACE?

I just read about polynomial-size circuit families and I have a question as the title. I know P/poly is defined as the class PSIZE of languages that have polynomial-size circuits. But what about other languages in other classes that have polynomial…
user2879534
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are all problems in P in NP

I know this is a similar question to this post, but I want to further clarify my understanding. In the picture from wikipedia: I understand that every problem that's in $P$ is also in NP. does this mean that everything from MergeSort, to linear…
Katie Melosto
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if $L \in NP$ then its mapping reducible to HALT?

This is true because every language in $NP$ is decidable and therefore HALTS but how do I formally show this?
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Is the certificate for primality testing polynomial in the length of the input?

If we were to assume that primality testing was in NP. What would the certificate be, so that a polynomial time verifier can check the number X is indeed prime?
WeCanBeFriends
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Do any decision problems exist outside NP and NP-Hard?

This question asks about a corner case of NP classes. From Wikipedia, NP is defined as: the set of all decision problems for which the instances where the answer is "yes" have efficiently verifiable proofs of the fact that the answer is indeed…
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NP problem solving

Prove that every NP problem is solvable in $O(2^{n^{k}})$ where $k$ is constant. My approach was that, let's look at the 3-SAT problem. We can solve it by bruteforce in $O(2^n)$, where $n$ is number of variables in 3-SAT. Also, we know that 3-SAT…
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Need help to proof that NP⊆NP4

NP4 = { L | There exists a non deterministic polynomial Turing machine M, such that for every x∈L, M accepts x on at least 4 paths in the computational tree of M on x. and for every x∉L,M accepts x on at most 3 paths in the computational tree of M…
Daniel
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