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From Wikipedia, the free encyclopedia
Unsolved problem in computer science
Unsolved problem in computer science :

^ A nondeterministic Turing machine can move to a state that is not determined by the previous state. Such a machine could solve an NP problem in polynomial time by falling into the correct answer state (by luck), then conventionally verifying it. Such machines are not practical for solving realistic problems but can be used as theoretical models.

^ Exactly how efficient a solution must be to pose a threat to cryptography depends on the details. A solution of



O
(

N

2


)


{\displaystyle O(N^{2})}

with a reasonable constant term would be disastrous. On the other hand, a solution that is



Ω
(

N

4


)


{\displaystyle \Omega (N^{4})}

in almost all cases would not pose an immediate practical danger.




^ Fortnow, Lance (2009). "The status of the P versus NP problem" (PDF) . Communications of the ACM . 52 (9): 78–86. CiteSeerX 10.1.1.156.767 . doi : 10.1145/1562164.1562186 . S2CID 5969255 . Archived from the original (PDF) on 24 February 2011 . Retrieved 26 January 2010 .

^ Fortnow, Lance (2013). The Golden Ticket: P, NP, and the Search for the Impossible . Princeton, NJ: Princeton University Press. ISBN 9780691156491 .

^ Cook, Stephen (1971). "The complexity of theorem proving procedures" . Proceedings of the Third Annual ACM Symposium on Theory of Computing . pp. 151–158. doi : 10.1145/800157.805047 . ISBN 9781450374644 . S2CID 7573663 .

^ L. A. Levin (1973). Универсальные задачи перебора [Problems of Information Transmission]. Пробл. передачи информ (in Russian). 9 (3): 115–116.

^ NSA (2012). "Letters from John Nash" (PDF) .

^ Hartmanis, Juris. "Gödel, von Neumann, and the P = NP problem" (PDF) . Bulletin of the European Association for Theoretical Computer Science . 38 : 101–107.

^ 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.

^ Jump up to: a b c William I. Gasarch (June 2002). "The P=?NP poll" (PDF) . SIGACT News . 33 (2): 34–47. CiteSeerX 10.1.1.172.1005 . doi : 10.1145/564585.564599 . S2CID 36828694 .

^ William I. Gasarch . "The Second P=?NP poll" (PDF) . SIGACT News . 74 .

^ Jump up to: a b "Guest Column: The Third P =? NP Poll1" (PDF) . Retrieved 25 May 2020 .

^ Scott Aaronson. "PHYS771 Lecture 6: P, NP, and Friends" . Retrieved 27 August 2007 .

^ "MSc course: Foundations of Computer Science" . www.cs.ox.ac.uk . Retrieved 25 May 2020 .

^ Colbourn, Charles J. (1984). "The complexity of completing partial Latin squares" . Discrete Applied Mathematics . 8 (1): 25–30. doi : 10.1016/0166-218X(84)90075-1 .

^ I. Holyer (1981). "The NP-completeness of some edge-partition problems". SIAM J. Comput . 10 (4): 713–717. doi : 10.1137/0210054 .

^ Aviezri Fraenkel and D. Lichtenstein (1981). "Computing a perfect strategy for n × n chess requires time exponential in n " . Journal of Combinatorial Theory, Series A . 31 (2): 199–214. doi : 10.1016/0097-3165(81)90016-9 .

^ David Eppstein . "Computational Complexity of Games and Puzzles" .

^ Fischer, Michael J. ; Rabin, Michael O. (1974). "Super-Exponential Complexity of Presburger Arithmetic" . Proceedings of the SIAM-AMS Symposium in Applied Mathematics . 7 : 27–41. Archived from the original on 15 September 2006 . Retrieved 15 October 2017 .

^ Valiant, Leslie G. (1979). "The complexity of enumeration and reliability problems". SIAM Journal on Computing . 8 (3): 410–421. doi : 10.1137/0208032 .

^ Jump up to: a b R. E. Ladner "On the structure of polynomial time reducibility," Journal of the ACM 22, pp. 151–171, 1975. Corollary 1.1. ACM site .

^ Arvind, Vikraman; Kurur, Piyush P. (2006). "Graph isomorphism is in SPP" . Information and Computation . 204 (5): 835–852. doi : 10.1016/j.ic.2006.02.002 .

^ Schöning, Uwe (1988). "Graph isomorphism is in the low hierarchy" . Journal of Computer and System Sciences . 37 (3): 312–323. doi : 10.1016/0022-0000(88)90010-4 .

^ Babai, László (2018). "Group, graphs, algorithms: the graph isomorphism problem". Proceedings of the International Congress of Mathematicians—Rio de Janeiro 2018. Vol. IV. Invited lectures . World Sci. Publ., Hackensack, NJ. pp. 3319–3336. MR 3966534 .

^ Lance Fortnow . Computational Complexity Blog: Complexity Class of the Week: Factoring . 13 September 2002.

^ Pisinger, D. 2003. "Where are the hard knapsack problems?" Technical Report 2003/08, Department of Computer Science, University of Copenhagen, Copenhagen, Denmark

^ Kawarabayashi, K. I.; Kobayashi, Y.; Reed, B. (2012). "The disjoint paths problem in quadratic time" . Journal of Combinatorial Theory, Series B . 102 (2): 424–435. doi : 10.1016/j.jctb.2011.07.004 .

^ Johnson, David S. (1987). "The NP-completeness column: An ongoing guide (edition 19)". Journal of Algorithms . 8 (2): 285–303. CiteSeerX 10.1.1.114.3864 . doi : 10.1016/0196-6774(87)90043-5 .

^ Gondzio, Jacek; Terlaky, Tamás (1996). "3 A computational view of interior point methods" . In J. E. Beasley (ed.). Advances in linear and integer programming . Oxford Lecture Series in Mathematics and its Applications. Vol. 4. New York: Oxford University Press. pp. 103–144. MR 1438311 . Postscript file at website of Gondzio and at McMaster University website of Terlaky .

^ Jump up to: a b Cook, Stephen (April 2000). "The P versus NP Problem" (PDF) . Clay Mathematics Institute . Retrieved 18 October 2006 .

^ Rosenberger, Jack (May 2012). "P vs. NP poll results" . Communications of the ACM . 55 (5): 10.

^ Scott Aaronson (4 September 2006). "Reasons to believe" . , point 9.

^ See Horie, S.; Watanabe, O. (1997). "Hard instance generation for SAT". Algorithms and Computation . Lecture Notes in Computer Science. Vol. 1350. Springer. pp. 22–31. arXiv : cs/9809117 . Bibcode : 1998cs........9117H . doi : 10.1007/3-540-63890-3_4 . ISBN 978-3-540-63890-2 . for a reduction of factoring to SAT. A 512 bit factoring problem (8400 MIPS-years when factored) translates to a SAT problem of 63,652 variables and 406,860 clauses.

^ See, for example, Massacci, F. & Marraro, L. (2000). "Logical cryptanalysis as a SAT problem". Journal of Automated Reasoning . 24 (1): 165–203. CiteSeerX 10.1.1.104.962 . doi : 10.1023/A:1006326723002 . S2CID 3114247 . in which an instance of DES is encoded as a SAT problem with 10336 variables and 61935 clauses. A 3DES problem instance would be about 3 times this size.

^ De, Debapratim; Kumarasubramanian, Abishek; Venkatesan, Ramarathnam (2007). "Inversion attacks on secure hash functions using SAT solvers". Theory and Applications of Satisfiability Testing--SAT 2007 . International Conference on Theory and Applications of Satisfiability Testing. Springer. pp. 377–382. doi : 10.1007/978-3-540-72788-0_36 .

^ Berger B , Leighton T (1998). "Protein folding in the hydrophobic-hydrophilic (HP) model is NP-complete". J. Comput. Biol . 5 (1): 27–40. CiteSeerX 10.1.1.139.5547 . doi : 10.1089/cmb.1998.5.27 . PMID 9541869 .

^ History of this letter and its translation from Michael Sipser. "The History and Status of the P versus NP question" (PDF) .

^ David S. Johnson (2012). "A Brief History of NP-Completeness, 1954–2012". CiteSeerX 10.1.1.399.1480 . From pages 359–376 of Optimization Stories, M. Grötschel (editor), a special issue of ¨ Documenta Mathematica, published in August 2012 and distributed to attendees at the 21st International Symposium on Mathematical Programming in Berlin.

^ Knuth, Donald E. (20 May 2014). Twenty Questions for Donald Knuth . informit.com . InformIT . Retrieved 20 July 2014 .

^ L. R. Foulds (October 1983). "The Heuristic Problem-Solving Approach". Journal of the Operational Research Society . 34 (10): 927–934. doi : 10.2307/2580891 . JSTOR 2580891 .

^ R. Impagliazzo, "A personal view of average-case complexity," sct, pp.134, 10th Annual Structure in Complexity Theory Conference (SCT'95), 1995

^ "Tentative program for the workshop on "Complexity and Cryptography: Status of Impagliazzo's Worlds" " . Archived from the original on 15 November 2013.

^ T. P. Baker; J. Gill; R. Solovay. (1975). "Relativizations of the P =? NP Question". SIAM Journal on Computing . 4 (4): 431–442. doi : 10.1137/0204037 .

^ Razborov, Alexander A.; Steven Rudich (1997). "Natural proofs" . Journal of Computer and System Sciences . 55 (1): 24–35. doi : 10.1006/jcss.1997.1494 .

^ S. Aaronson & A. Wigderson (2008). Algebrization: A New Barrier in Complexity Theory (PDF) . Proceedings of ACM STOC'2008. pp. 731–740. doi : 10.1145/1374376.1374481 .

^ Aaronson, Scott . "Is P Versus NP Formally Independent?" (PDF) . .

^ Ben-David, Shai; Halevi, Shai (1992). On the independence of P versus NP . Technion (Technical report). Vol. 714. Archived from the original (GZIP) on 2 March 2012. .

^ John Markoff (8 October 2009). "Prizes Aside, the P-NP Puzzler Has Consequences" . The New York Times .

^ Gerhard J. Woeginger . "The P-versus-NP page" . Retrieved 24 June 2018 .

^ Markoff, John (16 August 2010). "Step 1: Post Elusive Proof. Step 2: Watch Fireworks" . The New York Times . Retrieved 20 September 2010 .

^ Elvira Mayordomo. "P versus NP" Archived 16 February 2012 at the Wayback Machine Monografías de la Real Academia de Ciencias de Zaragoza 26: 57–68 (2004).

^ Agrawal, Manindra; Kayal, Neeraj; Saxena, Nitin (2004). "PRIMES is in P" (PDF) . Annals of Mathematics . 160 (2): 781–793. doi : 10.4007/annals.2004.160.781 . JSTOR 3597229 .

^ Geere, Duncan (26 April 2012). " 'Travelling Salesman' movie considers the repercussions if P equals NP" . Wired UK . Retrieved 26 April 2012 .

^ Hardesty, Larry. "Explained: P vs. NP" .

^ Shadia, Ajam. "What is the P vs. NP problem? Why is it important?" .

^ Gasarch, William (7 October 2013). "P vs NP is Elementary? No— P vs NP is ON Elementary" . blog.computationalcomplexity.org . Retrieved 6 July 2018 .

^ Kirkpatrick, Noel (4 October 2013). "Elementary Solve for X Review: Sines of Murder" . TV.com . Retrieved 6 July 2018 .


Wikiquote has quotations related to P versus NP problem
If the solution to a problem is easy to check for correctness, must the problem be easy to solve?
The P versus NP problem is a major unsolved problem in theoretical computer science . In informal terms, it asks whether every problem whose solution can be quickly verified can also be quickly solved.

The informal term quickly , used above, means the existence of an algorithm solving the task that runs in polynomial time , such that the time to complete the task varies as a polynomial function on the size of the input to the algorithm (as opposed to, say, exponential time ). The general class of questions for which some algorithm can provide an answer in polynomial time is " P " or " class 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 is possible to verify the answer quickly. The class of questions for which an answer can be verified in polynomial time is NP , which stands for "nondeterministic polynomial time". [Note 1]

An answer to the P versus NP question would determine whether problems that can be verified in polynomial time can also be solved in polynomial time. If it turns out that P ≠ NP, which is widely believed, it would mean that there are problems in NP 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.

The problem has been called the most important open problem in computer science . [1] Aside from being an important problem in computational theory , a proof either way would have profound implications for mathematics, cryptography , algorithm research, artificial intelligence , game theory , multimedia processing, philosophy , economics and many other fields. [2]

It is one of the seven Millennium Prize Problems selected by the Clay Mathematics Institute , each of which carries a US$1,000,000 prize for the first correct solution.

Consider Sudoku , a game where the player is given a partially filled-in grid of numbers and attempts to complete the grid following certain rules. Given an incomplete Sudoku grid, of any size, is there at least one legal solution? Any proposed solution is easily verified, and the time to check a solution grows slowly (polynomially) as the grid gets bigger. However, all known algorithms for finding solutions take, for difficult examples, time that grows exponentially as the grid gets bigger. So, Sudoku is in NP (quickly checkable) but does not seem to be in P (quickly solvable). Thousands of other problems seem similar, in that they are fast to check but slow to solve. Researchers have shown that many of the problems in NP have the extra property that a fast solution to any one of them could be used to build a quick solution to any other problem in NP, a property called NP-completeness . Decades of searching have not yielded a fast solution to any of these problems, so most scientists suspect that none of these problems can be solved quickly. This, however, has never been proven.

The precise statement of the P versus NP problem was introduced in 1971 by Stephen Cook in his seminal paper "The complexity of theorem proving procedures" [3] (and independently by Leonid Levin in 1973 [4] ).

Although the P versus NP problem was formally defined in 1971, there were previous inklings of the problems involved, the difficulty of proof, and the potential consequences. In 1955, mathematician John Nash wrote a letter to the NSA , in which he speculated that cracking a sufficiently complex code would require time exponential in the length of the key. [5] If proved (and Nash was suitably skeptical) this would imply what is now called P ≠ NP, since a proposed key can easily be verified in polynomial time. Another mention of the underlying problem occurred in a 1956 letter written by Kurt Gödel to John von Neumann . Gödel asked whether theorem-proving (now known to be co-NP-complete ) could be solved in quadratic or linear time , [6] and pointed out one of the most important consequences –that if so, then the discovery of mathematical proofs could be automated.

The relation between the complexity classes 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 m
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