Post by be left blank===Polynomial-time algorithm===
An [[algorithm]] version of the subset sum problem is: given a set of
''N'' numbers ''x''<sub>1</sub>, ''x''<sub>2</sub>, ..., ''x<sub>N</
sub>'' and a number ''s'', output
* yes, if a subset sums to ''s'';
* no, if no subset sums to a number between {{nowrap|(1 −
''c'')''s''}} and ''s'' for small {{nowrap|''c'' > 0;}}
* an answer, if a subset sums to a number between {{nowrap|(1 −
''c'')''s''}} and ''s'' but no subset sums to ''s''.
If all numbers are non-negative, subset sum is solvable in time
polynomial in ''N'' and 1/''c''.
The solution to subset sum provides the solution to the original
subset sum problem with small (nonnegative numbers). If any sum of the
numbers is specified by at most ''P'' bits, the solution
{{nowrap|''c'' {{=}} 2<sup>−''P''</sup>}} is equivalent to the
solution of the problem exactly. The polynomial time algorithm for
subset sum is an exact algorithm with running time polynomial in ''N''
and 2<sup>''P''</sup> (i.e., exponential in ''P'').
<code> initialize a list ''S'' to contain one element 0.
<code> for each ''i'' from 1 to ''N'' do
<code> let ''T'' be a list of ''x<sub>i</sub>'' + ''y'', for all
''y'' in ''S''
<code> let ''U'' be the union of ''T'' and ''S''
<code> sort ''U''
<code> make ''S'' empty
<code> let ''y'' be the smallest element of ''U''
<code> add ''y'' to ''S''
<code> for each element ''z'' of ''U'' in increasing order do
<code> //trim the list by eliminating numbers close to one
another
<code> //throw out elements greater than ''s''
<code> if ''y'' + ''cs''/''N'' < ''z'' ≤ ''s'', set ''y'' = ''z''
add ''z'' to ''S''
<code> if ''S'' contains a number between (1 − ''c'')''s'' and ''s'',
output ''yes'', otherwise ''no''</code>
This algorithm is polynomial time because lists ''S'', ''T'' and ''U''
always remain of size polynomial in ''N'' and 1/''c'' and, as long as
they are of polynomial size, all operations on them can be done in
polynomial time. The size of lists is kept polynomial by a trimming
step, only includes a number ''z'' into ''S'' if it is greater than
the previous one by ''cs''/''N'' and not greater than ''s''.
This step ensures each element in ''S'' is smaller than the next by at
least ''cs''/''N'' and does not contain elements greater than ''s''. A
property list consists of no more than ''N''/''c'' elements.
The algorithm is correct because each step introduces an additive
error of at most ''cs''/''N'' and ''N'' steps together introduce the
error of at most ''cs''.
XXXCAPCAPCAPDDDreprogram:http://en.wikipedia.org/wiki/
P_versus_NP_problem http://meami.org/search.php
MODE/CODE MODE.CODE
<MODE>
{{pp-move-indef}}
{{Millennium Problems}}
[[File:Complexity classes.svg|thumb|250px|Diagram of complexity
classes provided that '''P''' ≠ '''NP'''. The existence of problems
outside both '''P''' and '''NP'''-complete in this case was
established by [[NP-intermediate|Ladner's theorem]].<ref
name="Ladner75">R. E. Ladner "On the structure of polynomial time
reducibility," J.ACM, 22, pp. 151–171, 1975. Corollary 1.1. [http://
portal.acm.org/citation.cfm?
id=321877&dl=ACM&coll=&CFID=15151515&CFTOKEN=6184618 ACM site].</
ref>]]
The '''P versus NP problem''' is a major [[List of unsolved problems
in computer science|unsolved problem in computer science]].
Informally, it asks whether every problem whose solution can be
efficiently checked by a computer can also be efficiently ''solved''
by a computer. It was introduced in 1971 by [[Stephen Cook]] in his
paper "The complexity of theorem proving procedures"<ref>{{Cite book|
last=Cook|first=Stephen|authorlink=Stephen Cook|year=1971|chapter=The
complexity of theorem proving procedures|chapterurl=http://
portal.acm.org/citation.cfm?coll=GUIDE&dl=GUIDE&id=805047|
title=Proceedings of the Third Annual ACM Symposium on Theory of
Computing|pages=151–158}}</ref> and is considered by many to be the
most important open problem in the field.<ref>[[Lance Fortnow]],
[http://www.cs.uchicago.edu/~fortnow/papers/pnp-cacm.pdf ''The status
of the P versus NP problem''], Communications of the ACM 52 (2009),
no. 9, pp. 78–86. {{doi|10.1145/1562164.1562186}}</ref> It
is one of the seven [[Millennium Prize Problems]] selected by the
[[Clay Mathematics Institute]] to carry a US$ 1,000,000 prize for the
first correct solution.
In essence, the question '''P''' = '''NP'''? asks:
{{quote|Suppose that solutions to a problem can be verified quickly.
Then, can the solutions themselves also be computed quickly?}}
The theoretical notion of ''quick'' used here is an algorithm that
runs in [[polynomial time]]. The general class of questions for which
some algorithm can provide an answer in polynomial time is called
"class P" or just "'''[[P (complexity)|P]]'''".
For some questions, there is no known way to find an answer quickly,
but if one is provided with information showing what the answer is, it
may be possible to verify the answer quickly. The class of questions
for which an answer can be verified in polynomial time is called
'''[[NP (complexity)|NP]]'''.
Consider the [[subset sum problem]], an example of a problem that is
easy to verify, but whose answer may be difficult to compute. Given a
set of [[integer]]s, does some nonempty [[subset]] of them sum to 0?
For instance, does a subset of the set {{nowrap| {−2, −3, 15, 14, 7,
−10} }} add up to 0? The answer "yes, because {{nowrap| {−2, −3, −10,
15} }} add up to zero" can be quickly verified with three additions.
However, there is no known algorithm to find such a subset in
polynomial time (there is, however, in [[exponential time]], which
consists of 2<sup>''n''</sup>-1 tries), and indeed such an algorithm
cannot exist if the two complexity classes are not the same; hence
this problem is in '''NP''' (quickly checkable) but not necessarily in
'''P''' (quickly solvable).
An answer to the '''P''' = '''NP''' question would determine
whether problems like the subset-sum problem that can be verified in
polynomial time can also be solved in polynomial time. If it turned
out that '''P''' does not equal '''NP''', it would mean that there are
problems in '''NP''' (such as [[NP-complete]] problems) that are
harder to compute than to verify: they could not be solved in
polynomial time, but the answer could be verified in polynomial time.
==Context==
The relation between the [[complexity class]]es '''P''' and '''NP'''
is studied in [[computational complexity theory]], the part of the
[[theory of computation]] dealing with the resources required during
computation to solve a given problem. The most common resources are
time (how many steps it takes to solve a problem) and space (how much
memory it takes to solve a problem).
In such analysis, a model of the computer for which time must be
analyzed is required. Typically such models assume that the computer
is [[Deterministic computation|''deterministic'']] (given the
computer's present state and any inputs, there is only one possible
action that the computer might take) and ''sequential'' (it performs
actions one after the other).
In this theory, the class '''P''' consists of all those ''[[decision
problem]]s'' (defined [[#Formal definitions for P and NP|below]]) that
can be solved on a deterministic sequential machine in an amount of
time that is [[polynomial]] in the size of the input; the class
'''[[NP (complexity)|NP]]''' consists of all those decision problems
whose positive solutions can be verified in [[polynomial time]] given
the right information, or equivalently, whose solution can be found in
polynomial time on a [[Non-deterministic Turing machine|non-
deterministic]] machine.<ref>Sipser, Michael: ''Introduction to the
Theory of Computation, Second Edition, International Edition'', page
270. Thomson Course Technology, 2006. Definition 7.19 and Theorem
7.20.</ref> Clearly, '''P''' ⊆ '''NP'''. Arguably the biggest open
question in [[theoretical computer science]] concerns the relationship
between those two classes:
:Is '''P''' equal to '''NP'''?
In a 2002 poll of 100 researchers, 61 believed the answer to be no, 9
believed the answer is yes, and 22 were unsure; 8 believed the
question may be [[independent (mathematical logic)|independent]] of
the currently accepted axioms and so impossible to prove or
disprove.<ref name="poll">{{Cite journal|author=William I. Gasarch|
title=The P=?NP poll.|journal=SIGACT News|volume=33|issue=2|pages=34–
47|month=June | year=2002|url=http://www.cs.umd.edu/~gasarch/papers/
poll.pdf|doi=10.1145/1052796.1052804|format=PDF|accessdate=2008-12-29}}
</ref>
==NP-complete==
[[File:P np np-complete np-hard.svg|thumb|300px|right|[[Euler
diagram]] for [[P (complexity)|P]], [[NP (complexity)|NP]], NP-
complete, and NP-hard set of problems]]
{{Main|NP-complete}}
To attack the '''P''' = '''NP''' question the concept of '''NP'''-
completeness is very useful. Informally the '''NP'''-complete problems
are the "toughest" problems in '''NP''' in the sense that they are the
ones most likely not to be in '''P'''. '''NP'''-complete problems are
a set of problems that any other '''NP'''-problem can be reduced to in
polynomial time, but retain the ability to have their solution
verified in polynomial time. In comparison, [[NP-hard|'''NP'''-hard]]
problems are those at least as hard as '''NP'''-complete problems,
meaning all '''NP'''-problems can be reduced to them, but not all
'''NP'''-hard problems are in '''NP''', meaning not all of them have
solutions verifiable in polynomial time.
For instance, the decision problem version of the [[travelling
salesman problem]] is '''NP'''-complete, so ''any'' instance of
''any'' problem in '''NP''' can be transformed mechanically into an
instance of the traveling salesman problem, in polynomial time. The
traveling salesman problem is one of many such '''NP'''-complete
problems. If any '''NP'''-complete problem is in '''P''', then it
would follow that '''P''' = '''NP'''. Unfortunately, many important
problems have been shown to be '''NP'''-complete, and as of 2011 not a
single fast algorithm for any of them is known.
Based on the definition alone it's not obvious that '''NP'''-complete
problems exist. A trivial and contrived '''NP'''-complete problem can
be formulated as: given a description of a Turing machine M guaranteed
to halt in polynomial time, does there exist a polynomial-size input
that M will accept?<ref name="Scott">{{Cite web|author=Scott Aaronson|
title=PHYS771 Lecture 6: P, NP, and Friends|url=http://
www.scottaaronson.com/democritus/lec6.html |accessdate=2007-08-27}}</
ref> It is in '''NP''' because (given an input) it is simple to check
whether or not M accepts the input by simulating M; it is '''NP'''-
complete because the verifier for any particular instance of a problem
in '''NP''' can be encoded as a polynomial-time machine M that takes
the solution to be verified as input. Then the question of whether the
instance is a yes or no instance is determined by whether a valid
input exists.
The first natural problem proven to be '''NP'''-complete was the
[[Boolean satisfiability problem]]. This result
came to be known as [[Cook–Levin theorem]]; its
proof that satisfiability is NP-complete contains technical details
about Turing machines as they relate to the definition of '''NP'''.
However, after this problem was proved to be NP-complete, [[reduction
(complexity)|proof by reduction]] provided a simpler way to show that
many other problems are in this class. Thus, a vast class of seemingly
unrelated problems are all reducible to one another, and are in a
sense "the same problem".
==Harder problems==
{{See also|Complexity class}}
Although it is unknown whether '''P''' = '''NP''', problems outside of
'''P''' are known. A number of succinct problems (problems that
operate not on normal input, but on a computational description of the
input) are known to be [[EXPTIME#EXPTIME-complete|'''EXPTIME'''-
complete]]. Because it can be shown that '''P''' <math>\subsetneq</
math> '''[[EXPTIME]]''', these problems are outside '''P''', and so
require more than polynomial time. In fact, by the [[time hierarchy
theorem]], they cannot be solved in significantly less than
exponential time. Examples include finding a perfect strategy for
chess (on an N×N board)<ref name="Fraenkel1981">{{Cite journal| author
= [[Aviezri Fraenkel]] and D. Lichtenstein| title = Computing a
perfect strategy for n×n chess requires time exponential in n| journal
= J. Comb. Th. A| issue = 31| year = 1981| pages = 199–214}}</ref> and
some other board games.<ref>{{Cite web|title=Computational Complexity
of Games and Puzzles |url=http://www.ics.uci.edu/~eppstein/cgt/
hard.html |author=David Eppstein}}</ref>
The problem of deciding the truth of a statement in [[Presburger
arithmetic]] requires even more time. Fischer and [[Michael O. Rabin|
Rabin]] proved in 1974 that every algorithm that decides the truth of
Presburger statements has a runtime of at least <math>2^{2^{cn}}</
math> for some constant ''c''. Here, ''n'' is the length of the
Presburger statement. Hence, the problem is known to need more than
exponential run time. Even more difficult are the [[List of
undecidable problems|undecidable problems]], such as the [[halting
problem]]. They cannot be completely solved by any algorithm, in the
sense that for any particular algorithm there is at least one input
for which that algorithm will not produce the right answer; it will
either produce the wrong answer, finish without giving a conclusive
answer, or otherwise run forever without producing any answer at all.
==Problems in NP not known to be in P or NP-complete==
It was shown by Ladner that if P ≠ NP then there exist
problems in NP that are neither in P nor NP-complete.<ref
name="Ladner75" /> Such problems are called [[NP-intermediate]]
problems. The [[graph isomorphism problem]], the [[discrete logarithm
problem]] and the [[integer factorization problem]] are examples of
problems believed to be NP-intermediate. They are some of the very few
NP problems not known to be in P or to be NP-complete.
The [[graph isomorphism problem]] is the computational problem of
determining whether two finite [[Graph (mathematics)|graph]]s are
[[graph isomorphism|isomorphic]]. An important unsolved problem in
complexity theory is whether the graph isomorphism problem is in P, NP-
complete, or NP-intermediate. The answer is not known, but it is
believed that the problem is at least not NP-complete.<ref
name="AK06">{{cite journal
| first1 = Vikraman
| last1 = Arvind
| first2 = Piyush P.
| last2 = Kurur
| title = Graph isomorphism is in SPP
| journal = Information and Computation
| volume = 204
| issue = 5
| year = 2006
| pages = 835–852
| doi = 10.1016/j.ic.2006.02.002
| postscript = .}}</ref> If graph isomorphism is NP-complete, the
[[polynomial time hierarchy]] collapses to its second level.<ref>[[Uwe
Schöning]], "Graph isomorphism is in the low hierarchy", Proceedings
of the 4th Annual [[Symposium on Theoretical Aspects of Computer
Science]], 1987, 114–124; also: ''Journal of Computer and System
Sciences'', vol. 37 (1988), 312–323</ref> Since it is widely
believed that the polynomial hierarchy does not collapse to any finite
level, it is believed that graph isomorphism is not NP-complete. The
best algorithm for this problem, due to [[Laszlo Babai]] and [[Eugene
Luks]] has run time 2<sup>O(√(''n'' log ''n''))</sup> for
graphs with ''n'' vertices.
The [[integer factorization problem]] is the computational problem of
determining the [[prime factorization]] of a given integer. Phrased as
a decision problem, it is the problem of deciding whether the input
has a factor less than ''k''. No efficient integer factorization
algorithm is known, and this fact forms the basis of several modern
cryptographic systems, such as the [[RSA]] algorithm. The integer
factorization problem is in NP and in co-NP (and even in UP and co-
UP<ref>[[Lance Fortnow]]. Computational Complexity Blog: Complexity
Class of the Week: Factoring. September 13, 2002.
http://weblog.fortnow.com/2002/09/complexity-class-of-week-factoring.html</ref>).
If the problem is NP-complete, the polynomial time hierarchy will
collapse to its first level (i.e., NP will equal co-NP). The best
known algorithm for integer factorization is the [[general number
field sieve]], which takes expected time O(e<sup>(64/9)<sup>1/3</
sup>(''n''.log 2)<sup>1/3</sup>(log (''n''.log 2))<sup>2/3</sup></
sup>) to factor an ''n''-bit integer. However, the best known
[[quantum algorithm]] for this problem, [[Shor's algorithm]], does run
in polynomial time. Unfortunately, this fact doesn't say much about
where the problem lies with respect to non-quantum complexity classes.
==Does P mean "easy"?==
[[File:KnapsackEmpComplexity.GIF|thumb|310 px|The graph shows time
(average of 100 instances in msec using a 933 MHz Pentium III)
vs.problem size for knapsack problems for a state-of-the-art
specialized algorithm. Quadratic fit suggests that empirical
algorithmic complexity for instances with 50–10,000 variables is
O((log ''n'')<sup>2</sup>).<ref name=Pisinger2003>Pisinger, D.
2003. "Where are the hard knapsack problems?" Technical Report
2003/08, Department of Computer Science, University of Copenhagen,
Copenhagen, Denmark</ref>]]
All of the above discussion has assumed that '''P''' means "easy" and
"not in '''P'''" means "hard". This assumption, known as ''[[Cobham's
thesis]]'', though a common and reasonably accurate assumption in
complexity theory, is not always true in practice; the size of
constant factors or exponents may have practical importance, or there
may be solutions that work for situations encountered in practice
despite having poor worst-case performance in theory (this is the case
for instance for the [[simplex algorithm]] in [[linear programming]]).
Other solutions violate the Turing machine model on which '''P''' and
'''NP''' are defined by introducing concepts like randomness and
[[quantum computation]].
Because of these factors, even if a problem is shown to be NP-
complete, and even if '''P''' ≠ '''NP''', there may still be effective
approaches to tackling the problem in practice. There are algorithms
for many NP-complete problems, such as the [[knapsack problem]], the
[[travelling salesman problem]] and the [[boolean satisfiability
problem]], that can solve to optimality many real-world instances in
reasonable time. The empirical [[average-case complexity]] (time vs.
problem size) of such algorithms can be surprisingly low.
==Reasons to believe P ≠ NP==
According to a poll,<ref name="poll"/> many computer scientists
believe that '''P''' ≠ '''NP'''. A key reason for this
belief is that after decades of studying these problems no one has
been able to find a polynomial-time algorithm for any of more than
3000 important known '''NP'''-complete problems (see [[List of NP-
complete problems]]). These algorithms were sought long before the
concept of '''NP'''-completeness was even defined ([[Karp's 21 NP-
complete problems]], among the first found, were all well-known
existing problems at the time they were shown to be NP-complete).
Furthermore, the result '''P''' = '''NP''' would imply many other
startling results that are currently believed to be false, such as
'''NP''' = '''[[co-NP]]''' and '''P''' = [[PH (complexity)|'''PH''']].
It is also intuitively argued that the existence of problems that are
hard to solve but for which the solutions are easy to verify matches
real-world experience.<ref>{{Cite web|url=http://scottaaronson.com/
blog/?p=122 |author=Scott Aaronson |title=Reasons to believe}}, point
9.</ref>
{{quote|If P <nowiki>=</nowiki> NP, then the world would be
a profoundly different place than we usually assume it to be. There
would be no special value in "creative leaps," no fundamental gap
between solving a problem and recognizing the solution once it's
found. Everyone who could appreciate a symphony would be Mozart;
everyone who could follow a step-by-step argument would be Gauss...|
[[Scott Aaronson]], [[MIT]]}}
On the other hand, some researchers believe that there is
overconfidence in believing '''P''' ≠ '''NP''' and that researchers
should explore proofs of '''P''' = '''NP''' as well. For example, in
2002 these statements were made:<ref name="poll" />
{{quote|The main argument in favor of '''P''' ≠ '''NP''' is
the total lack of fundamental progress in the area of exhaustive
search. This is, in my opinion, a very weak argument. The space of
algorithms is very large and we are only at the beginning of its
exploration. [. . .] The resolution of [[Fermat's Last Theorem]] also
shows that very simple questions may be settled only by very deep
theories.|[[Moshe Y. Vardi]], [[Rice University]]}}
{{quote|Being attached to a speculation is not a good guide to
research planning. One should always try both directions of every
problem. Prejudice has caused famous mathematicians to fail to solve
famous problems whose solution was opposite to their expectations,
even though they had developed all the methods required.|[[Anil
Nerode]], [[Cornell University]]}}
==Consequences of proof==
One of the reasons the problem attracts so much attention is the
consequences of the answer. A proof that '''P''' = '''NP''' could have
stunning practical consequences, if the proof leads to efficient
methods for solving some of the important problems in NP. It is also
possible that a proof would not lead directly to efficient methods,
perhaps if the proof is non-constructive, or the size of the bounding
polynomial is too big to be efficient in practice. The consequences,
both positive and negative, arise since various NP-complete problems
are fundamental in many fields.
Cryptography, for example, relies on certain problems being difficult.
A constructive and efficient solution to an NP-complete problem such
as [[Boolean satisfiability problem#3-satisfiability|3-SAT]] would
break most existing cryptosystems including [[public-key
cryptography]], a foundation for many modern security applications
such as secure economic transactions over the Internet, and
[[symmetric cipher]]s such as [[Advanced Encryption Standard|AES]] or
[[Triple DES|3DES]], used for the encryption of communications data.
These would need to be modified or replaced by [[information-theoretic
security|information-theoretically secure]] solutions.
On the other hand, there are enormous positive consequences that would
follow from rendering tractable many currently mathematically
intractable problems. For instance, many problems in [[operations
research]] are NP-complete, such as some types of [[integer
programming]], and the [[travelling salesman problem]], to name two of
the most famous examples. Efficient solutions to these problems would
have enormous implications for [[logistics]]. Many other important
problems, such as some problems in [[protein structure prediction]],
are also '''NP'''-complete;<ref name="Berger">{{Cite journal|
author=Berger B, Leighton T |title=Protein folding in the hydrophobic-
hydrophilic (HP) model is NP-complete |journal=J. Comput. Biol. |
volume=5 |issue=1 |pages=27–40 |year=1998 |pmid=9541869 |doi=10.1089/
cmb.1998.5.27 }}</ref> if these problems were efficiently solvable it
could spur considerable advances in biology.
But such changes may pale in significance compared to the revolution
an efficient method for solving NP-complete problems would cause in
mathematics itself. According to [[Stephen Cook]],<ref name="Official
Problem Description">{{Cite journal|last=Cook|first=Stephen|
authorlink=Stephen Cook|title=The P versus NP Problem|publisher=[[Clay
Mathematics Institute]]|year=2000|month=April|url=http://
www.claymath.org/millennium/P_vs_NP/Official_Problem_Description.pdf|accessdate=2006-10-18}}</ref>
{{quote|...it would transform mathematics by allowing a computer to
find a formal proof of any theorem which has a proof of a reasonable
length, since formal proofs can easily be recognized in polynomial
time. Example problems may well include all of the [[Clay Math
Institute#Millennium Prize Problems|CMI prize problems]].}}
Research mathematicians spend their careers trying to prove theorems,
and some proofs have taken decades or even centuries to find after
problems have been stated—for instance, [[Fermat's Last Theorem]] took
over three centuries to prove. A method that is guaranteed to find
proofs to theorems, should one exist of a "reasonable" size, would
essentially end this struggle.
A proof that showed that '''P''' ≠ '''NP''' would lack the practical
computational benefits of a proof that '''P''' = '''NP''', but would
nevertheless represent a very significant advance in computational
complexity theory and provide guidance for future research. It would
allow one to show in a formal way that many common problems cannot be
solved efficiently, so that the attention of researchers can be
focused on partial solutions or solutions to other problems. Due to
widespread belief in '''P''' ≠ '''NP''', much of this focusing of
research has already taken place.<ref>{{Cite journal|title=The
Heuristic Problem-Solving Approach |author=L. R. Foulds |journal=The
Journal of the Operational Research Society |volume=34 |issue=10 |
month=October | year=1983 |pages=927–934 |url=http://www.jstor.org/pss/
2580891 |doi=10.2307/2580891}}</ref>
A "not equal" resolution to the P versus NP problem still leaves open
the [[average-case complexity]] of hard problems in NP. For example,
it is possible that SAT requires exponential time in the worst case,
but that almost all randomly selected instances of it are efficiently
solvable. [[Russell Impagliazzo]] has described five hypothetical
"worlds" that could result from different possible resolutions to the
average-case complexity question.<ref>R. Impagliazzo, [http://
cseweb.ucsd.edu/~russell/average.ps "A personal view of average-case
complexity,"] sct, pp.134, 10th Annual Structure in Complexity Theory
Conference (SCT'95), 1995</ref> These range from "Algorithmica",
where P=NP and problems like SAT can be solved efficiently in all
instances, to "Cryptomania", where P≠NP and generating hard instances
of problems outside P is easy, with three intermediate possibilities
reflecting different possible distributions of difficulty over
instances of NP-hard problems. The "world" where P≠NP but all
problems in NP are tractable in the average case is called
"Heuristica" in the paper. A [[Princeton University]] workshop in
2009 studied the status of the five worlds.<ref>http://
intractability.princeton.edu/blog/2009/05/program-for-workshop-on-
impagliazzos-worlds/</ref>
==Results about difficulty of proof==
Although the '''P''' = '''NP'''? problem itself remains open, despite
a million-dollar prize and a huge amount of dedicated research,
efforts to solve the problem have led to several new techniques. In
particular, some of the most fruitful research related to the '''P'''
= '''NP''' problem has been in showing that existing proof techniques
are not powerful enough to answer the question, thus suggesting that
novel technical approaches are required.
As additional evidence for the difficulty of the problem, essentially
all known proof techniques in [[computational complexity]] theory fall
into one of the following classifications, each of which is known to
be insufficient to prove that '''P''' ≠ '''NP''':
{| class="wikitable"
|-
!Classification
!Definition
|-
|Relativizing proofs
|Imagine a world where every algorithm is allowed to make queries to
some fixed subroutine called an [[oracle machine|oracle]], and the
running time of the oracle is not counted against the running time of
the algorithm. Most proofs (especially classical ones) apply uniformly
in a world with oracles regardless of what the oracle does. These
proofs are called ''relativizing''. In 1975, Baker, Gill, and [[Robert
M. Solovay|Solovay]] showed that '''P''' = '''NP''' with respect to
some oracles, while '''P''' ≠ '''NP''' for other oracles.<ref>T. P.
Baker, J. Gill, R. Solovay. ''Relativizations of the P =? NP
Question''. [[SIAM Journal on Computing]], 4(4): 431–442 (1975)</ref>
Since relativizing proofs can only prove statements that are uniformly
true with respect to all possible oracles, this showed that
relativizing techniques cannot resolve '''P''' = '''NP'''.
|-
|Natural proofs
|In 1993, [[Alexander Razborov]] and [[Steven Rudich]] defined a
general class of proof techniques for circuit complexity lower bounds,
called ''[[natural proof]]s''. At the time all previously known
circuit lower bounds were natural, and circuit complexity was
considered a very promising approach for resolving '''P''' = '''NP'''.
However, Razborov and Rudich showed that, if [[one-way functions]]
exist, then no natural proof method can distinguish between '''P'''
and '''NP'''. Although one-way functions have never been formally
proven to exist, most mathematicians believe that they do, and a proof
or disproof of their existence would be a much stronger statement than
the quantification of '''P''' relative to '''NP'''. Thus it is
unlikely that natural proofs alone can resolve '''P''' = '''NP'''.
|-
|Algebrizing proofs
|After the Baker-Gill-Solovay result, new non-relativizing proof
techniques were successfully used to prove that [[IP (complexity)|IP]]
= [[PSPACE]]. However, in 2008, [[Scott Aaronson]] and [[Avi
Wigderson]] showed that the main technical tool used in the '''IP''' =
'''PSPACE''' proof, known as ''arithmetization'', was also
insufficient to resolve '''P''' = '''NP'''.<ref>S. Aaronson and A.
Wigderson. Algebrization: A New Barrier in Complexity Theory, in
Proceedings of ACM STOC'2008, pp. 731–740.</ref>
|}
These barriers are another reason why '''NP'''-complete problems are
useful: if a polynomial-time algorithm can be demonstrated for an
'''NP'''-complete problem, this would solve the '''P''' = '''NP'''
problem in a way not excluded by the above results.
These barriers have also led some computer scientists to suggest that
the P versus NP problem may be [[Independence (mathematical logic)|
independent]] of standard axiom systems like [[ZFC]] (cannot be proved
or disproved within them). The interpretation of an independence
result could be that either no polynomial-time algorithm exists for
any NP-complete problem, and such a proof cannot be constructed in
(e.g.) ZFC, or that polynomial-time algorithms for NP-complete
problems may exist, but it's impossible to prove in ZFC that such
algorithms are correct.<ref>{{Cite web|url=http://
www.scottaaronson.com/papers/pnp.pdf|first=Scott|last=Aaronson|authorlink=Scott
Aaronson|title=Is P Versus NP Formally Independent?|postscript=<!--
Bot inserted parameter. Either remove it; or change its value to "."
for the cite to end in a ".", as necessary. -->{{inconsistent
citations}}}}.</ref> However, if it can be shown, using techniques of
the sort that are currently known to be applicable, that the problem
cannot be decided even with much weaker assumptions extending the
[[Peano axioms]] (PA) for integer arithmetic, then
there would necessarily exist nearly-polynomial-time algorithms for
every problem in NP.<ref>{{Cite document|title=On the independence of
P versus NP|first1=Shai|last1=Ben-David|first2=Shai|last2=Halevi|
series=Technical Report|volume=714|publisher=Technion|year=1992|
url=http://www.cs.technion.ac.il/~shai/ph.ps.gz|postscript=<!-- Bot
inserted parameter. Either remove it; or change its value to "." for
the cite to end in a ".", as necessary. -->{{inconsistent
citations}}}}.</ref> Therefore, if one believes (as most complexity
theorists do) that not all problems in NP have efficient algorithms,
it would follow that proofs of independence using those techniques
cannot be possible. Additionally, this result implies that proving
independence from PA or ZFC using currently known techniques is no
easier than proving the existence of efficient algorithms for all
problems in NP.
==Claimed solutions <span id="Deolalikar"/>==
While the P versus NP problem is generally considered
unsolved,<ref>{{Cite web|author=[[John Markoff]]|url=http://
www.nytimes.com/2009/10/08/science/Wpolynom.html|title=Prizes Aside,
the P-NP Puzzler Has Consequences|work=[[The New York Times]]|date=8
October 2009}}</ref> many amateur and some professional researchers
have claimed solutions. Woeginger (2010) has a comprehensive
list.<ref>{{Cite web|title=The P-versus-NP page|url=http://
www.win.tue.nl/~gwoegi/P-versus-NP.htm|author=Gerhard J. Woeginger|
date=2010-08-09|accessdate=2010-08-12}}</ref> An August 2010 claim of
proof that P ≠ NP, by Vinay Deolalikar, researcher at [[HP Labs]],
[[Palo Alto]], received heavy Internet and press attention after being
initially described as "seem[ing] to be a relatively serious attempt"
by two leading specialists.<ref name="NYT2010">{{Cite news|
last=Markoff|first=John|title=Step 1: Post Elusive Proof. Step 2:
Watch Fireworks.|url=http://www.nytimes.com/2010/08/17/science/
17proof.html?_r=1|accessdate=20 September 2010|newspaper=The New York
Times|date=16 August 2010}}</ref> The proof has been reviewed
publicly by academics,<ref>{{Cite web
|url=http://michaelnielsen.org/polymath1/index.php?
title=Deolalikar_P_vs_NP_paper
|author=[[Polymath project]] wiki
|title=Deolalikar's P vs NP paper}}</ref><ref>Science News, [http://
www.sciencenews.org/index/generic/activity/view/id/63252/title/Crowdsourcing_peer_review
"Crowdsourcing peer review"]</ref> and [[Neil Immerman]], an expert in
the field, had pointed out two possibly fatal errors in the
proof.<ref>{{Cite web
|title=Fatal Flaws in Deolalikar's Proof?
|url=http://rjlipton.wordpress.com/2010/08/12/fatal-flaws-in-
deolalikars-proof/
|author=[[Richard J. Lipton|Dick Lipton]]
|date=12 August 2010
}}</ref>
As of September 15, 2010, Deolalikar was reported to be working on a
detailed expansion of his attempted proof.<ref>{{Cite web
|url=http://rjlipton.wordpress.com/2010/09/15/an-update-on-vinay-
deolalikars-proof/
|title=An Update on Vinay Deolalikar's Proof
|author=[[Richard J. Lipton|Dick Lipton]]
|date=15 September 2010
|accessdate=December 31, 2010
}}</ref> However, the general consensus amongst theoretical computer
scientists is now that the attempted proof is not correct, or even a
significant advancement in our understanding of the problem.
==Logical characterizations==
The '''P''' = '''NP''' problem can be restated in terms of expressible
certain classes of logical statements, as a result of work in
[[descriptive complexity]]. All languages (of finite structures with a
fixed [[signature (logic)|signature]] including a [[linear order]]
relation) in '''P''' can be expressed in [[first-order logic]] with
the addition of a suitable least [[fixed point combinator]]
(effectively, this, in combination with the order, allows the
definition of recursive functions); indeed, (as long as the signature
contains at least one predicate or function in addition to the
distinguished order relation [so that the amount of space taken to
store such finite structures is actually polynomial in the number of
elements in the structure]), this precisely characterizes '''P'''.
Similarly, '''NP''' is the set of languages expressible in existential
[[second-order logic]]—that is, second-order logic restricted to
exclude [[universal quantification]] over relations, functions, and
subsets. The languages in the [[polynomial hierarchy]], '''[[PH
(complexity)|PH]]''', correspond to all of second-order logic. Thus,
the question "is '''P''' a proper subset of '''NP'''" can be
reformulated as "is existential second-order logic able to describe
languages (of finite linearly ordered structures with nontrivial
signature) that first-order logic with least fixed point cannot?". The
word "existential" can even be dropped from the previous
characterization, since '''P''' = '''NP''' if and only if '''P''' =
'''PH''' (as the former would establish that '''NP''' = '''co-NP''',
which in turn implies that '''NP''' = '''PH'''). [[PSPACE]] =
[[NPSPACE]] as established [[Savitch's theorem]], this follows
directly from the fact that the square of a polynomial function is
still a polynomial function. However, it is believed, but not proven,
a similar relationship may not exist between the polynomial time
complexity classes, [[P (complexity)|P]] and [[NP (complexity)|NP]] so
the question is still open.
==Polynomial-time algorithms==
No algorithm for any '''NP'''-complete problem is known to run in
polynomial time. However, there are algorithms for '''NP'''-complete
problems with the property that if '''P''' = '''NP''', then the
algorithm runs in polynomial time (although with enormous constants,
making the algorithm impractical). The following algorithm, due to
[[Leonid Levin|Levin]],{{Citation needed|date=September 2010}} is such
an example. It correctly accepts the '''NP'''-complete language
[[subset sum problem|SUBSET-SUM]], and runs in polynomial time if and
only if '''P''' = '''NP''':
// Algorithm that accepts the NP-complete language SUBSET-SUM.
//
// This is a polynomial-time algorithm if and only if
'''P'''='''NP'''.
//
// "Polynomial-time" means it returns "yes" in polynomial time when
// the answer should be "yes", and runs forever when it is "no".
//
// Input: S = a finite set of integers
// Output: "yes" if any subset of S adds up to 0.
// Runs forever with no output otherwise.
// Note: "Program number P" is the program obtained by
// writing the integer P in binary, then
// considering that string of bits to be a
// program. Every possible program can be
// generated this way, though most do nothing
// because of syntax errors.<br />
FOR N = 1...infinity
FOR P = 1...N
Run program number P for N steps with input S
IF the program outputs a list of distinct integers
AND the integers are all in S
AND the integers sum to 0<br />
THEN
OUTPUT "yes" and HALT
If, and only if, '''P''' = '''NP''', then this is a polynomial-time
algorithm accepting an '''NP'''-complete language. "Accepting" means
it gives "yes" answers in polynomial time, but is allowed to run
forever when the answer is "no".
This algorithm is enormously impractical, even if '''P''' = '''NP'''.
If the shortest program that can solve SUBSET-SUM in polynomial time
is ''b'' bits long, the above algorithm will try 2<sup>b</sup>−1 other
programs first.
An [[algorithm]] version of the subset sum problem gives a set of
''N'' numbers ''x''<sub>1</sub>, ''x''<sub>2</sub>, ..., ''x<sub>N</
sub>'' and a number ''s'', outputs
* yes, if a subset sums to ''s'';
* no, if no subset sums to a number between {{nowrap|(1 −
''c'')''s''}} and ''s'' for small {{nowrap|''c'' > 0;}}
* an answer, if a subset sums to a number between {{nowrap|(1 −
''c'')''s''}} and ''s'' but no subset sums to ''s''.
If all numbers are non-negative, subset sum is solvable in polynomial-
time in ''N'' and 1/''c''.
The solution to subset sum provides the solution to the original
subset sum problem with small (nonnegative numbers). If any sum of the
numbers is specified by at most ''P'' bits, the solution
{{nowrap|''c'' {{=}} 2<sup>−''P''</sup>}} is equivalent to solving the
problem exactly. The polynomial-time algorithm for subset sum is an
exact algorithm with polynomial running time in ''N''
and 2<sup>''P''</sup> (i.e., exponential in ''P'').
// initializes a list ''S'' to contain one element 0.
// for each ''i'' from 1 to ''N'' do
// lets ''T'' be a list of ''x<sub>i</sub>'' + ''y'', for all
''y'' in ''S''
// lets ''U'' be the union of ''T'' and ''S''
// sorts ''U''
// makes ''S'' empty
// lets ''y'' be the smallest element of ''U''
// adds ''y'' to ''S''
// for each element ''z'' of ''U'' in increasing order do
// //trims list eliminate numbers close to one another
// //throws out elements greater than ''s''
// if ''y'' + ''cs''/''N'' < ''z'' ≤ ''s'', set ''y'' = ''z''
adds ''z'' to ''S''
// if ''S'' contains a number between (1 − ''c'')''s'' and ''s'',
outputs ''yes'', otherwise // ''no''
This algorithm is polynomial-time as lists ''S'', ''T'' and ''U''
remain polynomial size in ''N'' and 1/''c'' and as long as they are
polynomial size all operations on them can be done in polynomial-time.
The size of lists is kept polynomial by the trim step including a
number ''z'' into ''S'' if it is greater than the previous one by
''cs''/''N'' and not greater than ''s'' ensures each element in ''S''
is smaller than the next by at least ''cs''/''N'' and does not contain
elements greater than ''s''. A property list consists of ''N''/''c''
or less elements. The algorithm correctly introduces an additive error
of at most ''cs''/''N'' and ''N'' steps together introduce the error
of at most ''cs''.<ref name="Musatov">{{Cite web|author=M. Musatov|
title=Polynomial Time Algorithm in Wiki Code|url=http://
groups.google.com/group/comp.edu/browse_thread/thread/
1dca0eaeb513a249#|format=HTML|accessdate=2011-04-15}}</ref>
==Formal definitions for P and NP==
Conceptually a ''decision problem'' is a problem that takes as input
some [[String (computer science)|string]] ''w'' over an alphabet <math>
\Sigma</math>, and outputs "yes" or "no". If there is an [[algorithm]]
(say a [[Turing machine]], or a [[Computer programming|computer
program]] with unbounded memory) that can produce the correct answer
for any input string of length ''n'' in at most <math>c \cdot n^k + c</
math> steps, where ''k'' and ''c'' are constants independent of the
input string, then we say that the problem can be solved in
''polynomial time'' and we place it in the class '''P'''. Formally,
'''P''' is defined as the set of all languages that can be decided by
a deterministic polynomial-time Turing machine. That is,
'''P''' = <math>\{ L : L=L(M) \text{ for some deterministic polynomial-
time Turing machine } M \}</math>
where <math>L(M) = \{ w\in\Sigma^{*}: M \text{ accepts } w \}</math>
and a deterministic polynomial-time Turing machine is a deterministic
Turing machine ''M'' that satisfies the following two conditions:
# <math>M \text{ halts on all input } w</math>; and
# there exists <math>k \in N</math> such that <math>T_M(n)\in
O(n^{k})</math> (where O refers to the [[Big O notation#Formal
definition|big O notation]]),
::where <math>T_M(n) = \max\{ t_M(w) : w\in\Sigma^{*}, \left|w\right|
= n \}</math>
::and <math>t_M(w) = \text{ number of steps }M\text{ takes to halt on
input }w.</math>
'''NP''' can be defined similarly using nondeterministic Turing
machines (the traditional way). However, a modern approach to define
'''NP''' is to use the concept of ''[[Certificate (complexity)|
certificate]]'' and ''verifier''. Formally, '''NP''' is defined as the
set of languages over a finite alphabet that have a verifier that runs
in polynomial time, where the notion of "verifier" is defined as
follows.
Let ''L'' be a language over a finite alphabet, <math>\Sigma</math>.
<math>L\in\mathbf{NP}</math> if, and only if, there exists a binary
relation <math>R\subset\Sigma^{*}\times\Sigma^{*}</math> and a
positive integer ''k'' such that the following two conditions are
satisfied:
# For all <math>x\in\Sigma^{*}</math>, <math>x\in L \Leftrightarrow
\exists y\in\Sigma^{*}</math> such that <math>(x,y)\in R\;</math> and
<math>\left|y\right|\in O(\left|x\right|^{k})</math>; and
# the language <math>L_{R} = \{ x\# y:(x,y)\in R\}</math> over <math>
\Sigma\cup\{\#\}</math> is decidable by a Turing machine in polynomial
time.
A Turing machine that decides <math>L_{R}</math> is called a
''verifier'' for ''L'' and a ''y'' such that <math>(x,y)\in R</math>
is called a ''certificate of membership'' of ''x'' in ''L''.
In general, a verifier does not have to be polynomial-time. However,
for ''L'' to be in '''NP''', there must be a verifier that runs in
polynomial time.
===Example===
Let <math>\mathit{COMPOSITE} = \{x\in\mathbb N:x=pq \;\text{for
integers}\; p, q > 1 \}</math> and <math>R = \{(x,y)\in\mathbb N\times
\mathbb N: 1<y \leq \sqrt x\; ; \;y\; \text{divides}\; x\}.</math>
Clearly, the question of whether a given ''x'' is a [[Composite number|
composite]] is equivalent to the question of whether ''x'' is a member
of <math>\mathit{COMPOSITE}</math>. It can be shown that <math>
\mathit{COMPOSITE}\in\mathbf{NP}</math> by verifying that <math>
\mathit{COMPOSITE}</math> satisfies the above definition (if we
identify natural numbers with their binary representations).
<math>\mathit{COMPOSITE}</math> also happens to be in '''P'''.<ref
name="Agrawal">{{Cite web|author=M. Agrawal, N. Kayal, N. Saxena|
title=Primes is in P|url=http://www.cse.iitk.ac.in/users/manindra/
algebra/primality_v6.pdf|format=PDF|accessdate=2008-12-29}}</
ref><ref>[[AKS primality test]]</ref>
==Formal definition for NP-completeness==
There are many equivalent ways of describing '''NP'''-completeness.
Let <math>\ L</math> be a language over a finite alphabet <math>\
\Sigma</math>.
<math>\ L</math> is '''NP'''-complete if, and only if, the following
two conditions are satisfied:
# <math>L\in\mathbf{NP}</math>; and
# any <math>L^{'}\in\mathbf{NP}</math> is polynomial-time-reducible to
<math>\ L</math> (written as <math>L^{'}\leq_{p} L</math>), where
<math>L^{'}\leq_{p} L</math> if, and only if, the following two
conditions are satisfied:
## There exists <math>f : \Sigma^{*}\rightarrow\Sigma^{*}</math> such
that <math>
\forall w\in\Sigma^{*}(w\in L^{'}\Leftrightarrow f(w)\in L)</math>;
and
## there exists a polynomial-time Turing machine that halts with <math>
\ f(w)</math> on its tape on any input <math>\ w</math>.
==See also==
* [[Game complexity]]
* [[Unsolved problems in computer science]]
* [[Unsolved problems in mathematics]]
==Notes==
{{Reflist|colwidth=30em}}
==Further reading==
* {{cite doi|10.1007/3-540-10843-2_23}}
* {{cite book | last = Garey | first = Michael | title = Computers and
Intractability | publisher = W.H. Freeman | location = San Francisco |
year = 1979 | isbn = 0716710455 }}
* {{cite book | last = Goldreich | first = Oded | title = P, Np, and
Np-Completeness | publisher = Cambridge University Press | location =
Cambridge | year = 2010 | isbn = 9780521122542 }}
* {{cite doi|10.1145/800061.808765}}
* {{cite book | last = Cormen | first = Thomas | title = Introduction
to Algorithms | publisher = MIT Press | location = Cambridge | year =
2001 | isbn = 0262032937 }}
* {{cite book | last = Papadimitriou | first = Christos | title =
Computational Complexity | publisher = Addison-Wesley | location =
Boston | year = 1994 | isbn = 0201530821 }}
* {{cite doi|10.1145/1562164.1562186}}
==External links==
{{Wikinews|1=P=NP math problem reported solved}}
* [http://www.claymath.org/millennium/ The Clay Mathematics Institute
Millennium Prize Problems]
* {{PDF|[http://www.claymath.org/millennium/P_vs_NP/
Official_Problem_Description.pdf The Clay Math Institute Official
Problem Description]|118 KB}}
* [http://www.claymath.org/Popular_Lectures/Minesweeper/ Ian Stewart
on Minesweeper as '''NP'''-complete at The Clay Math Institute]
* Gerhard J. Woeginger. [http://www.win.tue.nl/~gwoegi/P-versus-NP.htm
The P-versus-NP page]. A list of links to a number of purported
solutions to the problem. Some of these links state that P equals NP,
some of them state the opposite. It is probable that all these alleged
solutions are incorrect.
* [http://www.ics.uci.edu/~eppstein/cgt/hard.html Computational
Complexity of Games and Puzzles]
* {{CZoo|Class P|P#p}}, {{CZoo|Class NP|N#np}}
* [[Scott Aaronson]] [http://scottaaronson.com/blog/?p=122 's Shtetl
Optimized blog: Reasons to believe], a list of justifications for the
belief that P ≠ NP
{{ComplexityClasses}}
{{Use dmy dates|date=September 2010}}
{{DEFAULTSORT:P Versus Np Problem}}
[[Category:Structural complexity theory]]
[[Category:Mathematical optimization]]
[[Category:Conjectures]]
[[Category:Unsolved problems in mathematics]]
[[Category:Unsolved problems in computer science]]
[[Category:Millennium Prize Problems]]
[[ar:مسألة P=NP]]
[[ca:P versus NP]]
[[cs:Problém P versus NP]]
[[de:P-NP-Problem]]
[[es:Clases de complejidad P y NP]]
[[eo:Demando P = NP]]
[[fa:P = NP]]
[[fr:Problème P = NP]]
[[ko:P-NP 문제]]
[[it:Classi di complessità P e NP]]
[[ja:P≠NP予想]]
[[pt:P versus NP]]
[[ru:Равенство классов P и NP]]
[[simple:P versus NP]]
[[sr:П = НП проблем]]
[[fi:P=NP]]
[[sv:P=NP?]]
[[th:กลุ่มความซับซ้อน พี และ เอ็นพี]]
[[tr:P ile NP arasındaki ilişki]]
[[uk:Рівність класів P і NP]]
[[zh:P/NP问题]]
</CODE>