The class of decision problems for which a 'yes' answer can be verified in mP (that is, monotone polynomial-time). The monotonicity requirement applies only to the input bits, not to the bits that are guessed nondeterministically. So, in the corresponding circuit, one can have NOT gates so long as they depend only on the nondeterministic guess bits.
Defined in [GS90], where it was also shown that mNP is 'trivial': that is, it contains exactly the monotone problems in NP.
The definition of this class, due to [GS90], is not obvious. First, a monotone nondeterministic Turing machine is one such that, whenever it can make a transition with a 0 on its input tape, it can also make that same transition with a 1 on its input tape. (This restriction does not apply to the work tape.) A monotone alternating Turing machine is subject to the restriction that it can only reference an input bit x by, "there exists a z at most x," or "for all z at least x."
Then applying the result of [CKS81] that P = AL, mP is defined to be mAL: the class of decision problems solvable by a monotone alternating log-space Turing machine.
Actually there's a caveat: A monotone Turing machine or circuit can first negate the entire input, then perform a monotone computation. That way it becomes meaningful to talk about whether a monotone complexity class is closed under complement.
Strictly contained in mNP [Raz85].
Deciding whether a bipartite graph has a perfect matching, despite being both a monotone problem and in P, requires monotone circuits of superpolynomial size [Raz85b]. Letting MONO be the class of monotone problems, it follows that mP is strictly contained in MONO ∩ P.
See also: mNC1, mL, mNL, mcoNL.
The class of dashed hopes and idle dreams.
More formally: an "NP machine" is a nondeterministic polynomial-time Turing machine.
Then NP is the class of decision problems solvable by an NP machine such that
Equivalently, NP is the class of decision problems such that, if the answer is "yes," then there is a proof of this fact, of length polynomial in the size of the input, that can be verified in P (i.e. by a deterministic polynomial-time algorithm). On the other hand, if the answer is "no," then the algorithm must declare invalid any purported proof that the answer is "yes."
For example, the SAT problem is to decide whether a given Boolean formula has any satisfying truth assignments. SAT is in NP, since a "yes" answer can be proved by just exhibiting a satisfying assignment.
A decision problem is NP-complete if (1) it is in NP, and (2) any problem in NP can be reduced to it (under some notion of reduction). The class of NP-complete problems is sometimes called NPC.
That NP-complete problems exist is immediate from the definition. The seminal result of Cook [Coo71], Karp [Kar72], and Levin [Lev73] is that many natural problems (that have nothing to do with Turing machines) are NP-complete.
The first such problem to be shown NP-complete was SAT [Coo71]. Other classic NP-complete problems include:
For many, many more NP-complete problems, see [GJ79].
There exists an oracle relative to which P and NP are unequal [BGS75]. Indeed, P and NP are unequal relative to a random oracle with probability 1 [BG81] (see [AFM01] for a novel take on this result). Though random oracle results are not always indicative about the unrelativized case [CCG+94].
There even exists an oracle relative to which the P versus NP problem is outside the usual axioms of set theory [HH76].
If we restrict to monotone classes, mP is strictly contained in mNP [Raz85].
Perhaps the most important insight anyone has had into P versus NP is to be found in [RR97]. There the authors show that no 'natural proof' can separate P from NP (or more precisely, place NP outside of P/poly), unless secure pseudorandom generators do not exist. A proof is 'natural' if it satisfies two conditions called constructivity and largeness; essentially all lower bound techniques known to date satisfy these conditions. To obtain unnatural proof techniques, some people suspect we need to relate P versus NP to heavy-duty 'traditional' mathematics, for instance algebraic geometry. See [MS02] (and the survey article [Reg02]) for a development of this point of view.
For more on P versus NP (circa 1992) see [Sip92]. For an opinion poll, see [Gas02]. P vs NP is a "Millenium Prize" problem [CMI00].
If P equals NP, then NP equals its complement coNP. Whether NP equals coNP is also open. NP and coNP can be extended to the polynomial hierarchy PH.
The set of decision problems in NP, but not in P or NPC, is sometimes called NPI. If P does not equal NP then NPI is nonempty [Lad75].
Probabilistic generalizations of NP include MA and AM. If NP is in coAM (or BPP) then PH collapses to Σ2P [BHZ87].
PH also collapses to Σ2P if NP is in P/poly [KL82].
There exist oracles relative to which NP is not in BQP [BBB+97].
An alternate characterization is NP = PCP(log n, O(1)) [ALM+98].
Also, [Fag74] showed that NP is precisely the class of decision problems reducible to a graph-theoretic property expressible in second-order existential logic. This leads to the subclass SNP.
It is known that if any NP-complete language is sparse (contains no more than a polynomial number of strings of length ), then P = NP. [BH08] improved this result, showing that if any language in NP has an NP-hard set of subexponential density, then coNP is contained in NP/poly and thus, by [Yap82], PH collapses to the third level.
NP is equal to SO-E, the second-order queries where the second-order quantifiers are only existantials.
