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P (complexity)

Incomputational complexity theory,P, also known as PTIME orDTIME(nO(1)),

is one of the most fundamentalcomplexity classes. It contains alldecisionproblemsthat can be solved by adeterministic Turing machineusing apolynomialamount ofcomputation time, orpolynomial time.

Cobham's thesisholds that P is the class of computational problems that are"efficiently solvable" or "tractable"; in practice, some problems not known to be inP have practical solutions, and some that are in P do not, but this is a useful rule of

thumb.

Definition

A languageL is in P if and only if there exists a deterministic Turing machineM,such that

Mruns for polynomial time on all inputs For allx inL,Moutputs 1 For allx not inL,Moutputs 0

P can also be viewed as a uniform family ofboolean circuits. A languageL is in Pif and only if there exists apolynomial-time uniformfamily of boolean circuits

, such that

For all , takes n bits as input and outputs 1 bit For allx inL, For allx not inL,

The circuit definition can be weakened to use only alogspace uniformfamilywithout changing the complexity class.

Notable problems in P

P is known to contain many natural problems, including the decision versions of

linear programming, calculating thegreatest common divisor, and finding a

maximum matching. In 2002, it was shown that the problem of determining if anumber isprimeis in P.[1]The related class offunction problemsisFP.
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Several natural problems are complete forP, includingst-connectivity(or

reachability) on alternating graphs.[2]The article onP-complete problemslistsfurther relevant problems in P.

Relationships to other classes

A generalization ofP isNP, which is the class ofdecision problemsdecidable by a

non-deterministic Turing machinethat runs inpolynomial time. Equivalently, it isthe class of decision problems where each "yes" instance has a polynomial sizecertificate, and certificates can be checked by a polynomial time deterministic

Turing machine. The class of problems for which this is true for the "no" instances

is calledco-NP.P is trivially a subset ofNP and ofco-NP; most experts believe itis a strict subset,[3]although this (thePNP hypothesis) remains unproven.

Another open problem is whetherNP = co-NP (a negative answer would imply

PNP).

P is also known to be at least as large asL, the class of problems decidable in a

logarithmicamount ofmemory space. A decider using space cannot use

more than time, because this is the total number of possible

configurations; thus, L is a subset ofP. Another important problem is whetherL =P. We do know that P = AL, the set of problems solvable in logarithmic memory

byalternating Turing machines.P is also known to be no larger thanPSPACE, theclass of problems decidable in polynomial space. Again, whetherP = PSPACE isan open problem. To summarize:

Here,EXPTIMEis the class of problems solvable in exponential time. Of all theclasses shown above, only two strict containments are known:

P is strictly contained in EXPTIME. Consequently, all EXPTIME-hardproblems lie outside P, and at least one of the containments to the right ofPabove is strict (in fact, it is widely believed that all three are strict).

L is strictly contained in PSPACE.

The most difficult problems in P areP-completeproblems.

Another generalization ofP is P/poly, orNonuniform Polynomial-Time. If aproblem is in P/poly, then it can be solved in deterministic polynomial time

provided that anadvice stringis given that depends only on the length of the input.
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Unlike forNP, however, the polynomial-time machine doesn't need to detect

fraudulent advice strings; it is not a verifier. P/poly is a large class containingnearly all practical problems, including all ofBPP. If it contains NP, then the

polynomial hierarchycollapses to the second level. On the other hand, it also

contains some impractical problems, including someundecidable problemssuch asthe unary version of any undecidable problem.

In 1999, Jin-Yi Cai and D. Sivakumar, building on work byMitsunori Ogihara,showed that if there exists asparse languagethat is P-complete, then L = P.[4]

Properties

Polynomial-time algorithms are closed under composition. Intuitively, this says

that if one writes a function that is polynomial-time assuming that function calls

are constant-time, and if those called functions themselves require polynomialtime, then the entire algorithm takes polynomial time. One consequence of this isthat P islowfor itself. This is also one of the main reasons that P is considered to

be a machine-independent class; any machine "feature", such asrandom access,that can be simulated in polynomial time can simply be composed with the main

polynomial-time algorithm to reduce it to a polynomial-time algorithm on a morebasic machine.

Pure existence proofs of polynomial-time algorithms

Some problems are known to be solvable in polynomial-time, but no concretealgorithm is known for solving them. For example, theRobertsonSeymour

theoremguarantees that there is a finite list offorbidden minorsthat characterizes

(for example) the set of graphs that can be embedded on a torus; moreover,Robertson and Seymour showed that there is an O(n3) algorithm for determiningwhether a graph has a given graph as a minor. This yields anonconstructive proof

that there is a polynomial-time algorithm for determining if a given graph can beembedded on a torus, despite the fact that no concrete algorithm is known for this

problem.

Alternative characterizations

Indescriptive complexity,P can be described as the problems expressible inFO

(LFP), the class offirst-order logicwith aleast fixed pointoperator added to it. InImmerman's 1999 textbook on descriptive complexity,[5]Immerman ascribes this

result to Vardi[6]and to Immerman.[7]
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History

Kozen[8]states thatCobhamandEdmondsare "generally credited with the

invention of the notion of polynomial time." Cobham invented the class as a robustway of characterizing efficient algorithms, leading toCobham's thesis. However,

H. C. Pocklington, in a 1910 paper,[9][10]analyzed two algorithms for solvingquadratic congruences, and observed that one took time "proportional to a power of

the logarithm of the modulus" and contrasted this with one that took time

proportional "to the modulus itself or its square root", thus explicitly drawing adistinction between an algorithm that ran in polynomial time versus one that didnot.

NP (complexity)

From Wikipedia, the free encyclopedia

Jump to:navigation,search

List of unsolvedproblems in computer

science

Is P= NP?

Euler diagramforP, NP,NP-complete, andNP-hardset of problems. Theexistence of problems within NP but outside both P and NP-complete, under this

assumption, wasestablished by Ladner.[1]

Incomputational complexity theory,NP is one of the most fundamental

complexity classes. The abbreviation NP refers to "nondeterministicpolynomialtime."
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Intuitively, NP is the set of alldecision problemsfor which the instances where the

answer is "yes" have efficiently verifiable proofs of the fact that the answer isindeed "yes". More precisely, these proofs have to be verifiable inpolynomial time

by adeterministic Turing machine. In an equivalent formal definition, NP is the set

ofdecision problemswhere the "yes"-instances can be accepted inpolynomial timeby anon-deterministic Turing machine. The equivalence of the two definitions

follows from the fact that an algorithm on such a non-deterministic machineconsists of two phases, the first of which consists of a guess about the solution,

which is generated in a non-deterministic way, while the second consists of adeterministic algorithm that verifies or rejects the guess as a valid solution to the

problem.[2]

The complexity classPis contained in NP, but NP contains many important

problems, the hardest of which are calledNP-completeproblems, for which no

polynomial-timealgorithmsare known for solving them (although they can beverifiedin polynomial time). The most important open question in complexitytheory, theP = NP problem, asks whether polynomial time algorithms actually

exist forNP-complete, and by corollary, all NP problems. It is widely believed thatthis is not the case.[3]

Formal definition

The complexity class NP can be defined in terms ofNTIMEas follows:

Alternatively, NP can be defined using deterministic Turing machines as verifiers.AlanguageL is in NP if and only if there exist polynomialsp and q, and adeterministic Turing machineM, such that

For allx andy, the machineMruns in timep(|x|) on input (x,y) For allx inL, there exists a stringy of length q(|x|) such thatM(x,y) = 1 For allx not inL and all stringsy of length q(|x|),M(x,y) = 0

Introduction
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Many naturalcomputer scienceproblems are covered by the class NP. In

particular, the decision versions of many interestingsearch problemsandoptimization problems are contained in NP.

Verifier-based definition

In order to explain the verifier-based definition ofNP, let us consider thesubset

sum problem: Assume that we are given someintegers, such as {7, 3, 2, 5, 8},and we wish to know whether some of these integers sum up to zero. In this

example, the answer is "yes", since the subset of integers {3, 2, 5} corresponds

to the sum (3) + (2) + 5 = 0. The task of deciding whether such a subset withsum zero exists is called thesubset sum problem.

As the number of integers that we feed into the algorithm becomes larger, the

number of subsets grows exponentially, and in fact the subset sum problem is NP-complete. However, notice that, if we are given a particular subset (often called acertificate), we can easily check orverify whether the subset sum is zero, by just

summing up the integers of the subset. So if the sum is indeed zero, that particularsubset is theprooforwitnessfor the fact that the answer is "yes". An algorithm

that verifies whether a given subset has sum zero is called verifier. A problem is

said to be in NP if there exists a verifier for the problem that executes inpolynomial time. In case of the subset sum problem, the verifier needs onlypolynomial time, for which reason the subset sum problem is in NP.

The "no"-answer version of this problem is stated as: "given a finite set of integers,does every non-empty subset have a nonzero sum?". Note that the verifier-baseddefinition ofNP does notrequire an easy-to-verify certificate for the "no"-answers.

The class of problems with such certificates for the "no"-answers is calledco-NP.In fact, it is an open question whether all problems in NP also have certificates for

the "no"-answers and thus are in co-NP.

Machine-definition

Equivalent to the verifier-based definition is the following characterization: NP is

the set ofdecision problemssolvable by anon-deterministic Turing machinethatruns inpolynomial time. (This means that there is an accepting computation path ifa word is in the languageco-NPis defined dually with rejecting paths.) This

definition is equivalent to the verifier-based definition because a non-deterministicTuring machine could solve an NP problem in polynomial time by non-

deterministically selecting a certificate and running the verifier on the certificate.
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Similarly, if such a machine exists, then a polynomial time verifier can naturally beconstructed from it.

Examples

This is an incomplete list of problems that are in NP.

All problems inP(For, given a certificate for a problem in P, we can ignorethe certificate and just solve the problem in polynomial time. Alternatively,note that a deterministic Turing machine is also trivially a non-deterministic

Turing machine that just happens to not use any non-determinism.)

The decision problem version of theinteger factorization problem: givenintegers n and k, is there a factorfwith 1


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However, in practical uses, instead of spending computational resources looking

for an optimal solution, a good enough (but potentially suboptimal) solution mayoften be found in polynomial time. Also, the real life applications of some

problems are easier than their theoretical equivalents. For example, inputs to the

generalTravelling salesman problemneed not obey thetriangle inequality, unlikereal road networks.

Equivalence of definitions

The two definitions ofNP as the class of problems solvable by a nondeterministic

Turing machine(TM) in polynomial time and the class of problems verifiable by a

deterministic Turing machine in polynomial time are equivalent. The proof isdescribed by many textbooks, for example Sipser'sIntroduction to the Theory ofComputation, section 7.3.

To show this, first suppose we have a deterministic verifier. A nondeterministicmachine can simply nondeterministically run the verifier on all possible proof

strings (this requires only polynomially many steps because it cannondeterministically choose the next character in the proof string in each step, and

the length of the proof string must be polynomially bounded). If any proof is valid,

some path will accept; if no proof is valid, the string is not in the language and itwill reject.

Conversely, suppose we have a nondeterministic TM called A accepting a given

language L. At each of its polynomially many steps, the machine'scomputationtreebranches in at most a constant number of directions. There must be at least oneaccepting path, and the string describing this path is the proof supplied to the

verifier. The verifier can then deterministically simulate A, following only theaccepting path, and verifying that it accepts at the end. If A rejects the input, there

is no accepting path, and the verifier will never accept.

Relationship to other classes

NP contains all problems inP, since one can verify any instance of the problem by

simply ignoring the proof and solving it. NP is contained inPSPACEto showthis, it suffices to construct a PSPACE machine that loops over all proof stringsand feeds each one to a polynomial-time verifier. Since a polynomial-time machine

can only read polynomially many bits, it cannot use more than polynomial space,nor can it read a proof string occupying more than polynomial space (so we don't

have to consider proofs longer than this). NP is also contained inEXPTIME, sincethe same algorithm operates in exponential time.
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ThecomplementofNP,co-NP, contains those problems which have a simple

proof forno instances, sometimes called counterexamples. For example,primalitytestingtrivially lies in co-NP, since one can refute the primality of an integer by

merely supplying a nontrivial factor. NP and co-NP together form the first level inthepolynomial hierarchy, higher only than P.

NP is defined using only deterministic machines. If we permit the verifier to be

probabilistic (this however, is not necessarily a BPP machine[4]), we get the class

MA solvable using anArthur-Merlin protocolwith no communication from Merlinto Arthur.

NP is a class ofdecision problems; the analogous class of function problems isFNP.

Other characterizations

In terms ofdescriptive complexity theory,NP corresponds precisely to the set oflanguages definable by existentialsecond-order logic(Fagin's theorem).

NP can be seen as a very simple type ofinteractive proof system, where the prover

comes up with the proof certificate and the verifier is a deterministic polynomial-time machine that checks it. It is complete because the right proof string will make

it accept if there is one, and it is sound because the verifier cannot accept if there isno acceptable proof string.

A major result of complexity theory is that NP can be characterized as theproblems solvable byprobabilistically checkable proofswhere the verifier uses

O(log n) random bits and examines only a constant number of bits of the proofstring (the class PCP(log n, 1)). More informally, this means that the NP verifier

described above can be replaced with one that just "spot-checks" a few places in

the proof string, and using a limited number of coin flips can determine the correctanswer with high probability. This allows several results about the hardness ofapproximation algorithmsto be proven.

Example

The decision version of thetraveling salesman problemis in NP. Given an input

matrix of distances between n cities, the problem is to determine if there is a routevisiting all cities with total distance less than k.
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A proof certificate can simply be a list of the cities. Then verification can clearly

be done in polynomial time by a deterministic Turing machine. It simply adds thematrix entries corresponding to the paths between the cities.

A nondeterministic Turing machine can find such a route as follows:

At each city it visits it "guesses" the next city to visit, until it has visitedevery vertex. If it gets stuck, it stops immediately.

At the end it verifies that the route it has taken has cost less than kinO(n)time.

One can think of each guess as "forking" a new copy of the Turing machine tofollow each of the possible paths forward, and if at least one machine finds a route

of distance less than k, that machine accepts the input. (Equivalently, this can be

thought of as a single Turing machine that always guesses correctly)

Binary searchon the range of possible distances can convert the decision version

of Traveling Salesman to the optimization version, by calling the decision versionrepeatedly (a polynomial number of times).
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