View source for NP-completeness

{{Short description|Complexity class}}
{{confusing |date = July 2012}}
[[File:3SAT 17 svg.svg|thumb|400px|The [[Boolean satisfiability problem]] (SAT) asks to determine if a [[propositional formula]] (example depicted) can be made ''true'' by an appropriate assignment ("solution") of truth values to its variables. While it is easy to verify whether a given assignment renders the formula ''true'',<ref>For example, simply assigning ''true'' to each variable renders the 18th conjunct <math>\overline{m} \lor \overline{r} \lor \overline{s}</math> (and hence the complete formula) ''false''.</ref> no essentially faster method to find a satisfying assignment is known than to try all assignments in succession. [[Stephen Cook|Cook]] and [[Leonid Levin|Levin]] proved that each easy-to-verify problem can be solved as fast as SAT, which is hence NP-complete.]]

In [[computational complexity theory]], a problem is '''NP-complete''' when:
# It is a [[decision problem]], meaning that for any input to the problem, the output is either "yes" or "no".
# When the answer is "yes", this can be demonstrated through the existence of a short (polynomial length) ''solution''.
# The correctness of each solution can be verified quickly (namely, in [[polynomial time]]) and a [[brute-force search]] algorithm can find a solution by trying all possible solutions.
# The problem can be used to simulate every other problem for which we can verify quickly that a solution is correct. In this sense, NP-complete problems are the hardest of the problems to which solutions can be verified quickly. If we could find solutions of some NP-complete problem quickly, we could quickly find the solutions of every other problem to which a given solution can be easily verified.
 
The name "NP-complete" is short for "nondeterministic polynomial-time complete". In this name, "nondeterministic" refers to [[nondeterministic Turing machine]]s, a way of mathematically formalizing the idea of a brute-force search algorithm. [[Polynomial time]] refers to an amount of time that is considered "quick" for a [[deterministic algorithm]] to check a single solution, or for a nondeterministic Turing machine to perform the whole search. "[[Complete (complexity)|Complete]]" refers to the property of being able to simulate everything in the same [[complexity class]].

More precisely, each input to the problem should be associated with a set of solutions of polynomial length, the validity of each of which can be tested quickly (in [[polynomial time]]),<ref>{{Cite book| last=Cobham | first=Alan | author-link=Alan Cobham | year = 1965 | chapter = The intrinsic computational difficulty of functions | title = Proc. Logic, Methodology, and Philosophy of Science II | publisher = North Holland}}</ref> such that the output for any input is "yes" if the solution set is non-empty and "no" if it is empty. The complexity class of problems of this form is called [[NP (complexity)|NP]], an abbreviation for "nondeterministic polynomial time". A problem is said to be [[NP-hard]] if everything in NP can be transformed in polynomial time into it even though it may not be in NP. A problem is NP-complete if it is both in NP and NP-hard. The NP-complete problems represent the hardest problems in NP. If some NP-complete problem has a polynomial time algorithm, all problems in NP do. The set of NP-complete problems is often denoted by '''NP-C''' or '''NPC'''.

Although a solution to an NP-complete problem can be ''verified'' "quickly", there is no known way to ''find'' a solution quickly. That is, the time required to solve the problem using any currently known [[algorithm]] increases rapidly as the size of the problem grows. As a consequence, determining whether it is possible to solve these problems quickly, called the [[P versus NP problem]], is one of the fundamental [[List of open problems in computer science|unsolved problems in computer science]] today.

While a method for computing the solutions to NP-complete problems quickly remains undiscovered, [[computer scientist]]s and [[computer programmer|programmer]]s still frequently encounter NP-complete problems. NP-complete problems are often addressed by using [[heuristic (computer science)|heuristic]] methods and [[approximation algorithm]]s.

== Overview ==
NP-complete problems are in [[NP (complexity)|NP]], the set of all [[decision problem]]s whose solutions can be verified in polynomial time; ''NP'' may be equivalently defined as the set of decision problems that can be solved in polynomial time on a [[non-deterministic Turing machine]]. A problem ''p'' in NP is NP-complete if every other problem in NP can be transformed (or reduced) into ''p'' in polynomial time.{{cn|date=October 2022}}

It is not known whether every problem in NP can be quickly solved—this is called the [[P versus NP problem]]. But if ''any NP-complete problem'' can be solved quickly, then ''every problem in NP'' can, because the definition of an NP-complete problem states that every problem in NP must be quickly reducible to every NP-complete problem (that is, it can be reduced in polynomial time). Because of this, it is often said that NP-complete problems are ''harder'' or ''more difficult'' than NP problems in general.{{cn|date=October 2022}}

== Formal definition ==
{{See also |P %3D NP problem#NP-completeness |l1 = formal definition for NP-completeness (article ''P = NP'')}}

A decision problem <math>\scriptstyle C</math> is NP-complete if:{{cn|date=October 2022}}
# <math>\scriptstyle C</math> is in NP, and
# Every problem in NP is [[Many-one reduction|reducible]] to <math>\scriptstyle C</math> in polynomial time.<ref>{{cite book |author = J. van Leeuwen |year = 1998 |title = Handbook of Theoretical Computer Science |publisher = Elsevier |isbn = 978-0-262-72014-4 |page = 84}}</ref>

<math>\scriptstyle C</math> can be shown to be in NP by demonstrating that a candidate solution to <math>\scriptstyle C</math> can be verified in polynomial time.

Note that a problem satisfying condition 2 is said to be [[NP-hard]], whether or not it satisfies condition 1.<ref>{{cite book |author = J. van Leeuwen |year = 1998 |title = Handbook of Theoretical Computer Science |publisher = Elsevier |isbn = 978-0-262-72014-4 |page = 80}}</ref>

A consequence of this definition is that if we had a polynomial time algorithm (on a [[Universal Turing machine|UTM]], or any other [[Turing completeness|Turing-equivalent]] [[abstract machine]]) for <math>\scriptstyle C</math>, we could solve all problems in NP in polynomial time.

== Background ==
[[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]] sets of problems.  The left side is valid under the assumption that [[P versus NP problem|P≠NP]], while the right side is valid under the assumption that P=NP (except that the empty language and its complement are never NP-complete, and in general, not every problem in P or NP is NP-complete).]]

The concept of NP-completeness was introduced in 1971 (see [[Cook–Levin theorem]]), though the term ''NP-complete'' was introduced later. At the 1971 [[Symposium on Theory of Computing|STOC]] conference, there was a fierce debate between the computer scientists about whether NP-complete problems could be solved in polynomial time on a [[deterministic]] [[Turing machine]]. [[John Hopcroft]] brought everyone at the conference to a consensus that the question of whether NP-complete problems are solvable in polynomial time should be put off to be solved at some later date, since nobody had any formal proofs for their claims one way or the other.{{cn|date=August 2024}}  This is known as "the question of whether P=NP".

Nobody has yet been able to determine conclusively whether NP-complete problems are in fact solvable in polynomial time, making this one of the great [[unsolved problems of mathematics]].  The [[Clay Mathematics Institute]] is offering a US$1 million reward ([[Millennium Prize Problems|Millennium Prize]]) to anyone who has a formal proof that P=NP or that P≠NP.<ref>{{Cite web |last=Kiersz |first=Andy |title=An eminent mathematician claims to have solved one of math's greatest mysteries — and it's one of 6 problems with a $1 million prize |url=https://www.businessinsider.com/millennium-prize-problems-million-dollar-prize-2017-12 |access-date=2023-04-24 |website=Business Insider |language=en-US}}</ref>

The existence of NP-complete problems is not obvious. The [[Cook–Levin theorem]] states that the [[Boolean satisfiability problem]] is NP-complete, thus establishing that such problems do exist. In 1972, [[Richard Karp]] proved that several other problems were also NP-complete (see [[Karp's 21 NP-complete problems]]); thus, there is a class of NP-complete problems (besides the Boolean satisfiability problem). Since the original results, thousands of other problems have been shown to be NP-complete by reductions from other problems previously shown to be NP-complete; many of these problems are collected in {{harvtxt|Garey|Johnson|1979}}.

== NP-complete problems ==
[[File:Relative NPC chart.svg|thumb|300px|right|Some NP-complete problems, indicating the [[reduction (complexity)|reductions]] typically used to prove their NP-completeness]]
{{Main|List of NP-complete problems}}

The easiest way to prove that some new problem is NP-complete is first to prove that it is in NP, and then to reduce some known NP-complete problem to it. Therefore, it is useful to know a variety of NP-complete problems. The list below contains some well-known problems that are NP-complete when expressed as decision problems.
{{div col|colwidth=25em}}
* [[Boolean satisfiability problem|Boolean satisfiability problem (SAT)]]
* [[Knapsack problem]]
* [[Hamiltonian path problem]]
* [[Travelling salesman problem]] (decision version)
* [[Subgraph isomorphism problem]]
* [[Subset sum problem]]
* [[Clique problem]]
* [[Vertex cover problem]]
* [[Independent set problem]]
* [[Dominating set problem]]
* [[Graph coloring problem]]
* [[Sudoku]]
{{div col end}}

To the right is a diagram of some of the problems and the [[reduction (complexity)|reductions]] typically used to prove their NP-completeness. In this diagram, problems are reduced from bottom to top. Note that this diagram is misleading as a description of the mathematical relationship between these problems, as there exists a [[polynomial-time reduction]] between any two NP-complete problems; but it indicates where demonstrating this polynomial-time reduction has been easiest.

There is often only a small difference between a problem in P and an NP-complete problem. For example, the [[3-satisfiability]] problem, a restriction of the Boolean satisfiability problem, remains NP-complete, whereas the slightly more restricted [[2-satisfiability]] problem is in P (specifically, it is [[NL-complete]]), but the slightly more general max. 2-sat. problem is again NP-complete. Determining whether a graph can be colored with 2 colors is in P, but with 3 colors is NP-complete, even when restricted to [[planar graph]]s. Determining if a graph is a [[cycle graph|cycle]] or is [[bipartite graph|bipartite]] is very easy (in [[L (complexity)|L]]), but finding a maximum bipartite or a maximum cycle subgraph is NP-complete. A solution of the [[knapsack problem]] within any fixed percentage of the optimal solution can be computed in polynomial time, but finding the optimal solution is NP-complete.

===Intermediate problems===

An interesting example is the [[graph isomorphism problem]], the [[graph theory]] problem of determining whether a [[graph isomorphism]] exists between two graphs.  Two graphs are [[isomorphic]] if one can be [[isomorphism|transformed]] into the other simply by renaming [[vertex (graph theory)|vertices]]. Consider these two problems:
* Graph Isomorphism: Is graph G<sub>1</sub> isomorphic to graph G<sub>2</sub>?
* Subgraph Isomorphism: Is graph G<sub>1</sub> isomorphic to a subgraph of graph G<sub>2</sub>?

The Subgraph Isomorphism problem is NP-complete. The graph isomorphism problem is suspected to be neither in P nor NP-complete, though it is in NP. This is an example of a problem that is thought to be ''hard'', but is not thought to be NP-complete. This class is called ''NP-Intermediate problems'' and exists if and only if P≠NP.

== Solving NP-complete problems ==
At present, all known algorithms for NP-complete problems require time that is [[superpolynomial]] in the input size. The [[vertex cover]] problem has <math>O(1.2738^k + nk)</math><ref>{{Cite journal |last1=Chen |first1=Jianer |last2=Kanj |first2=Iyad A. |last3=Xia |first3=Ge |date=2010-09-06 |title=Improved upper bounds for vertex cover |journal=Theoretical Computer Science |volume=411 |issue=40 |pages=3736–3756 |doi=10.1016/j.tcs.2010.06.026 |issn=0304-3975|doi-access=free }}</ref> for some <math>k>0</math> and it is unknown whether there are any faster algorithms.

The following techniques can be applied to solve computational problems in general, and they often give rise to substantially faster algorithms:
* [[Approximation algorithm|Approximation]]: Instead of searching for an optimal solution, search for a solution that is at most a factor from an optimal one.
* [[Randomized algorithm|Randomization]]: Use randomness to get a faster average [[running time]], and allow the algorithm to fail with some small probability. Note: The [[Monte Carlo method]] is not an example of an efficient algorithm in this specific sense, although evolutionary approaches like [[Genetic algorithm]]s may be.
* Restriction: By restricting the structure of the input (e.g., to planar graphs), faster algorithms are usually possible.
* [[Parameterized complexity|Parameterization]]: Often there are fast algorithms if certain parameters of the input are fixed.
* [[Heuristic (computer science)|Heuristic]]: An algorithm that works "reasonably well" in many cases, but for which there is no proof that it is both always fast and always produces a good result.  [[Metaheuristic]] approaches are often used.

One example of a heuristic algorithm is a suboptimal <math>O(n\log n)</math> [[greedy coloring|greedy coloring algorithm]] used for [[graph coloring problem|graph coloring]] during the [[register allocation]] phase of some compilers, a technique called [[Register allocation#Graph-coloring allocation|graph-coloring global register allocation]]. Each vertex is a variable, edges are drawn between variables which are being used at the same time, and colors indicate the register assigned to each variable. Because most [[RISC]] machines have a fairly large number of general-purpose registers, even a heuristic approach is effective for this application.

== Completeness under different types of reduction ==
In the definition of NP-complete given above, the term ''reduction'' was used in the technical meaning of a polynomial-time [[many-one reduction]].

Another type of reduction is polynomial-time [[Turing reduction]].  A problem <math>\scriptstyle X</math> is polynomial-time Turing-reducible to a problem <math>\scriptstyle Y</math> if, given a subroutine that solves <math>\scriptstyle Y</math> in polynomial time, one could write a program that calls this subroutine and solves <math>\scriptstyle X</math> in polynomial time.  This contrasts with many-one reducibility, which has the restriction that the program can only call the subroutine once, and the return value of the subroutine must be the return value of the program.

If one defines the analogue to NP-complete with Turing reductions instead of  many-one reductions, the resulting set of problems won't be smaller than NP-complete; it is an open question whether it will be any larger.

Another type of reduction that is also often used to define NP-completeness is the [[logarithmic-space many-one reduction]] which is a many-one reduction that can be computed with only a logarithmic amount of space. Since every computation that can be done in [[logarithmic space]] can also be done in polynomial time it follows that if there is a logarithmic-space many-one reduction then there is also a polynomial-time many-one reduction. This type of reduction is more refined than the more usual polynomial-time many-one reductions and it allows us to distinguish more classes such as [[P-complete]]. Whether under these types of reductions the definition of NP-complete changes is still an open problem. All currently known NP-complete problems are NP-complete under log space reductions. All currently known NP-complete problems remain NP-complete even under much weaker reductions such as <math>AC_0</math> reductions and <math>NC_0</math> reductions. Some NP-Complete problems such as SAT are known to be complete even under polylogarithmic time projections.<ref>{{Cite journal | doi=10.1006/jcss.1998.1583 | last1=Agrawal | first1=M. | author1-link=Manindra Agrawal | last2=Allender | first2=E. | last3=Rudich | first3=Steven | author3-link=Steven Rudich | title=Reductions in Circuit Complexity: An Isomorphism Theorem and a Gap Theorem | year=1998 | journal=Journal of Computer and System Sciences | issn=1090-2724 | volume=57 | issue=2 | pages=127–143 | doi-access=free }}</ref> It is known, however, that [[AC0|AC<sup>0</sup>]] reductions define a strictly smaller class than polynomial-time reductions.<ref>{{Cite journal | last1=Agrawal | first1=M. | author1-link=Manindra Agrawal | last2=Allender | first2=E. | last3=Impagliazzo | first3=R. | last4=Pitassi | first4=T. | author4-link = Toniann Pitassi | last5=Rudich | first5=Steven | author5-link=Steven Rudich | title=Reducing the complexity of reductions | doi=10.1007/s00037-001-8191-1 | year=2001 | journal=Computational Complexity | issn=1016-3328 | volume=10 | pages=117–138 | issue=2 | s2cid=29017219 }}
</ref>

== Naming ==
According to [[Donald Knuth]], the name "NP-complete" was popularized by [[Alfred Aho]], [[John Hopcroft]] and [[Jeffrey Ullman]] in their celebrated textbook "The Design and Analysis of Computer Algorithms".  He reports that they introduced the change in the [[galley proofs]] for the book (from "polynomially-complete"), in accordance with the results of a poll he had conducted of the [[theoretical computer science]] community.<ref>[[Don Knuth]], Tracy Larrabee, and Paul M. Roberts, ''[http://tex.loria.fr/typographie/mathwriting.pdf Mathematical Writing] {{Webarchive|url=https://web.archive.org/web/20100827044400/http://tex.loria.fr/typographie/mathwriting.pdf |date=2010-08-27 }}'' § 25, ''MAA Notes No. 14'', MAA, 1989 (also [[Stanford University|Stanford]] Technical Report, 1987).</ref> Other suggestions made in the poll<ref>{{Cite journal
|doi = 10.1145/1811129.1811130
|volume = 6
|issue = 1
|pages = 12–18
|last = Knuth
|first = D. F.
|title = A terminological proposal
|journal = SIGACT News
|year = 1974
|s2cid = 45313676
}}</ref> included  "[[Labours of Hercules|Herculean]]", "formidable", [[Kenneth Steiglitz|Steiglitz]]'s "hard-boiled" in honor of Cook, and Shen Lin's acronym "PET", which stood for "probably exponential time", but depending on which way the [[P versus NP problem]] went, could stand for "{{sic|hide=y|provably}} exponential time" or "previously exponential time".<ref>See the poll, or [http://www.cs.princeton.edu/~wayne/kleinberg-tardos/08np-complete-2x2.pdf] {{Webarchive|url=https://web.archive.org/web/20110607082034/http://www.cs.princeton.edu/~wayne/kleinberg-tardos/08np-complete-2x2.pdf|date=2011-06-07}}.</ref>

== Common misconceptions ==
The following misconceptions are frequent.<ref>{{Cite news|url=http://www.nature.com/news/2000/000113/full/news000113-10.html
|doi = 10.1038/news000113-10
|title = DNA computer helps travelling salesman
|first = Philip |last= Ball
|journal = Nature
|year = 2000
}}</ref> 
* ''"NP-complete problems are the most difficult known problems."'' Since NP-complete problems are in NP, their running time is at most exponential. However, some problems have been proven to require more time, for example [[Presburger arithmetic]]. Of some problems, it has even been proven that they can never be solved at all, for example the [[halting problem]].
* ''"NP-complete problems are difficult because there are so many different solutions."'' On the one hand, there are many problems that have a solution space just as large, but can be solved in polynomial time (for example [[minimum spanning tree]]). On the other hand, there are NP-problems with at most one solution that are NP-hard under randomized polynomial-time reduction (see [[Valiant–Vazirani theorem]]).
* ''"Solving NP-complete problems requires exponential time."'' First, this would imply P ≠ NP, which is still an unsolved question. Further, some NP-complete problems actually have algorithms running in superpolynomial, but subexponential time such as O(2<sup>{{sqrt|''n''}}</sup>''n''). For example, the [[Independent set problem|independent set]] and [[Dominating set problem|dominating set]] problems for [[planar graph]]s are NP-complete, but can be solved in subexponential time using the [[planar separator theorem]].<ref>{{harvtxt|Bern|1990}}; {{harvtxt|Deĭneko|Klinz|Woeginger|2006}}; {{harvtxt|Dorn|Penninkx|Bodlaender|Fomin|2005}}; {{harvtxt|Lipton|Tarjan|1980}}.</ref>
* ''"Each instance of an NP-complete problem is difficult."'' Often some instances, or even most instances, may be easy to solve within polynomial time. However, unless P=NP, any polynomial-time algorithm must asymptotically be wrong on more than polynomially many of the exponentially many inputs of a certain size.<ref>{{Cite journal | last1 = Hemaspaandra | first1 = L. A. | last2 = Williams | first2 = R. | doi = 10.1145/2421119.2421135 | title = SIGACT News Complexity Theory Column 76 | journal = ACM SIGACT News | volume = 43 | issue = 4 | page = 70 | year = 2012 | s2cid = 13367514 }}</ref>
* ''"If P=NP, all cryptographic ciphers can be broken."'' A polynomial-time problem can be very difficult to solve in practice if the polynomial's degree or constants are large enough. In addition, [[information-theoretic security]] provides cryptographic methods that cannot be broken even with unlimited computing power.
* ''"A large-scale quantum computer would be able to efficiently solve NP-complete problems."'' The class of decision problems that can be efficiently solved (in principle) by a fault-tolerant quantum computer is known as BQP. However, BQP is not believed to contain all of NP, and if it does not, then it cannot contain any NP-complete problem.<ref>{{cite conference | last = Aaronson | first = Scott | editor-last = Schulman | editor-first = Leonard J. | contribution = BQP and the polynomial hierarchy | doi = 10.1145/1806689.1806711 | pages = 141–150 | publisher = Association for Computing Machinery | title = Proceedings of the 42nd ACM Symposium on Theory of Computing, STOC 2010, Cambridge, Massachusetts, USA, 5–8 June 2010 | year = 2010| arxiv = 0910.4698 | isbn = 978-1-4503-0050-6 }}</ref>

== Properties ==
Viewing a [[decision problem#Definition|decision problem]] as a formal language in some fixed encoding, the set NPC of all NP-complete problems is '''not closed''' under:
* [[Union (set theory)|union]]
* [[intersection]]
* [[concatenation]]
* [[Kleene star]]{{example needed|Give a counter example for each of the set operations.|date=April 2023}}

It is not known whether NPC is closed under [[Complement (complexity)|complementation]], since NPC=[[co-NP-complete|co-NPC]] if and only if NP=[[co-NP]], and since NP=co-NP is an [[Open problem|open question]].<ref>{{citation |title = Complexity and Cryptography: An Introduction |first1 = John |last1 = Talbot|first2=D. J. A.|last2 = Welsh |author2-link = Dominic Welsh |publisher = Cambridge University Press |year = 2006 |isbn = 9780521617710 |page=57 |url = https://books.google.com/books?id=y_ZwupY8pzUC&pg=PA57 |quote = The question of whether NP and co-NP are equal is probably the second most important open problem in complexity theory, after the P versus NP question.}}</ref>

== See also ==
* [[Almost complete]]
* [[Gadget (computer science)]]
* [[Ladner's theorem]]
* [[List of NP-complete problems]]
* [[NP-hard]]
* [[P = NP problem]]
* [[Strongly NP-complete]]
* [[Travelling Salesman (2012 film)]]

== References ==
=== Citations ===
{{Reflist|30em}}

=== Sources ===
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 |last2 = Penninkx |first2 = Eelko
 |last3 = Bodlaender |first3 = Hans L. | author3-link = Hans L. Bodlaender
 |last4 = Fomin |first4 = Fedor V.
 |doi = 10.1007/11561071_11
 |pages = 95–106
 |publisher = Springer-Verlag
 |series = Lecture Notes in Computer Science
 |title = Proc. 13th European Symposium on Algorithms (ESA '05)
 |volume = 3669
 |year = 2005
 |chapter = Efficient Exact Algorithms on Planar Graphs: Exploiting Sphere Cut Branch Decompositions
 |isbn = 978-3-540-29118-3
 }}.
* {{Cite journal
 |last1 = Lipton |first1 = Richard J. |author1-link = Richard J. Lipton
 |last2 = Tarjan |first2 = Robert E. |author2-link = Robert Tarjan
 |doi = 10.1137/0209046
 |journal = [[SIAM Journal on Computing]]
 |pages = 615–627
 |title = Applications of a planar separator theorem
 |volume = 9
 |year = 1980
 |issue = 3
 |s2cid = 12961628 }}.
{{refend}}

== Further reading ==
{{refbegin}}
* [[Scott Aaronson]], ''[https://arxiv.org/abs/quant-ph/0502072 NP-complete Problems and Physical Reality]'', ACM [[SIGACT]] News, Vol. 36, No. 1. (March 2005), pp.&nbsp;30–52.
* [[Lance Fortnow]], ''[http://people.cs.uchicago.edu/~fortnow/papers/pnp-cacm.pdf The status of the P versus NP problem]'', [[Commun. ACM]], Vol. 52, No. 9. (2009), pp.&nbsp;78–86.
{{refend}}

{{ComplexityClasses}}

{{DEFAULTSORT:Np-Complete}}
[[Category:1971 in computing]]
[[Category:NP-complete problems| ]]
[[Category:Complexity classes]]
[[Category:Mathematical optimization]]