FKT algorithm - Revision history

JJMC89 bot III: Moving :Category:Algorithms in graph theory to :Category:Graph algorithms per Wikipedia:Categories for discussion/Log/2024 October 4

2024-10-12T19:55:35Z

Moving Category:Algorithms in graph theory to Category:Graph algorithms per Wikipedia:Categories for discussion/Log/2024 October 4

← Previous revision		Revision as of 19:55, 12 October 2024
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	*More history, information, and examples can be found in chapter 2 and section 5.3.2 of Dmitry Kamenetsky's [https://digitalcollections.anu.edu.au/bitstream/1885/49338/2/02whole.pdf PhD thesis]		*More history, information, and examples can be found in chapter 2 and section 5.3.2 of Dmitry Kamenetsky's [https://digitalcollections.anu.edu.au/bitstream/1885/49338/2/02whole.pdf PhD thesis]

	[[Category:~~Algorithms~~ ~~in graph theory~~]]		[[Category:Graph algorithms]]
	[[Category:Planar graphs]]		[[Category:Planar graphs]]

JJMC89 bot III: Moving :Category:Graph algorithms to :Category:Algorithms in graph theory per Wikipedia:Categories for discussion/Log/2024 October 4#Category:Graph algorithms

2024-10-12T02:22:49Z

Moving Category:Graph algorithms to Category:Algorithms in graph theory per Wikipedia:Categories for discussion/Log/2024 October 4#Category:Graph algorithms

← Previous revision		Revision as of 02:22, 12 October 2024
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	*More history, information, and examples can be found in chapter 2 and section 5.3.2 of Dmitry Kamenetsky's [https://digitalcollections.anu.edu.au/bitstream/1885/49338/2/02whole.pdf PhD thesis]		*More history, information, and examples can be found in chapter 2 and section 5.3.2 of Dmitry Kamenetsky's [https://digitalcollections.anu.edu.au/bitstream/1885/49338/2/02whole.pdf PhD thesis]

	[[Category:~~Graph~~ ~~algorithms~~]]		[[Category:Algorithms in graph theory]]
	[[Category:Planar graphs]]		[[Category:Planar graphs]]

David Eppstein: /* Generalizations */ arxiv=

2024-08-24T21:18:53Z

Generalizations: arxiv=

← Previous revision		Revision as of 21:18, 24 August 2024
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	\| last1 = Thilikos \| first1 = Dimitrios M.		\| last1 = Thilikos \| first1 = Dimitrios M.
	\| last2 = Wiederrecht \| first2 = Sebastian		\| last2 = Wiederrecht \| first2 = Sebastian
			\| arxiv = 2207.04923
	\| doi = 10.1145/3664648		\| doi = 10.1145/3664648
	\| issue = 4		\| issue = 4

David Eppstein: /* Generalizations */ killing a vortex (avoiding the COI that would be entailed on further expanding some of the intermediate research in this direction)

2024-08-24T07:20:58Z

Generalizations: killing a vortex (avoiding the COI that would be entailed on further expanding some of the intermediate research in this direction)

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 | hdl= 1813/6700
 | hdl-access= free
  | last = Jerrum | first = Mark | author-link = Mark Jerrum
  | doi = 10.1007/BF01010403

David Eppstein: standardize on cs1

2024-08-23T23:29:43Z

standardize on cs1

← Previous revision		Revision as of 23:29, 23 August 2024
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	{{short description\|Algorithm for counting perfect matchings in planar graphs}}		{{short description\|Algorithm for counting perfect matchings in planar graphs}}
			{{CS1 config\|mode=cs1}}
	The '''Fisher–Kasteleyn–Temperley''' ('''FKT''') '''algorithm''', named after [[Michael Fisher]], [[Pieter Kasteleyn]], and [[Harold Neville Vazeille Temperley\|Neville Temperley]], counts the number of [[perfect matching]]s in a [[planar graph\|planar]] graph in polynomial time. This same task is [[Sharp-P-complete\|#P-complete]] for general graphs. For [[Matching (graph theory)\|matching]]s that are not required to be perfect, counting them remains #P-complete even for planar graphs. The key idea of the FKT algorithm is to convert the problem into a [[Pfaffian]] computation of a [[skew-symmetric matrix]] derived from a planar embedding of the graph. The Pfaffian of this matrix is then computed efficiently using standard [[Determinant#Algorithmic implementation\|determinant algorithms]].		The '''Fisher–Kasteleyn–Temperley''' ('''FKT''') '''algorithm''', named after [[Michael Fisher]], [[Pieter Kasteleyn]], and [[Harold Neville Vazeille Temperley\|Neville Temperley]], counts the number of [[perfect matching]]s in a [[planar graph\|planar]] graph in polynomial time. This same task is [[Sharp-P-complete\|#P-complete]] for general graphs. For [[Matching (graph theory)\|matching]]s that are not required to be perfect, counting them remains #P-complete even for planar graphs. The key idea of the FKT algorithm is to convert the problem into a [[Pfaffian]] computation of a [[skew-symmetric matrix]] derived from a planar embedding of the graph. The Pfaffian of this matrix is then computed efficiently using standard [[Determinant#Algorithmic implementation\|determinant algorithms]].

David Eppstein: replace conf version with journal version fixing CS1 errors along the way

2024-03-30T21:12:37Z

replace conf version with journal version fixing CS1 errors along the way

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 ==Applications==
-The FKT algorithm has seen extensive use in [[holographic algorithm]]s on planar graphs via [[matchgates]].<ref name=CLX/> For example, consider the planar version of the ice model mentioned above, which has the technical name #[[planar graph|PL]]-3-NAE-[[Boolean satisfiability problem|SAT]] (where NAE stands for "not all equal"). Valiant found a polynomial time algorithm for this problem which uses matchgates.<ref>{{cite conference
+The FKT algorithm has seen extensive use in [[holographic algorithm]]s on planar graphs via [[matchgates]].<ref name=CLX/> For example, consider the planar version of the ice model mentioned above, which has the technical name #[[planar graph|PL]]-3-NAE-[[Boolean satisfiability problem|SAT]] (where NAE stands for "not all equal"). Valiant found a polynomial time algorithm for this problem which uses matchgates.<ref>{{cite journal
- |year           = 2004
+ | mr = 2386281
 ==References==

Citation bot: Altered author-link. | Use this bot. Report bugs. | Suggested by Kline | Category:CS1 errors: access-date without URL | #UCB_Category 25/34

2024-03-30T20:56:25Z

Altered author-link. | Use this bot. Report bugs. | Suggested by Kline | Category:CS1 errors: access-date without URL | #UCB_Category 25/34

← Previous revision		Revision as of 20:56, 30 March 2024
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	\| volume = 27 \| issue = 12 \| year = 1961 \| pages = 1209–1225		\| volume = 27 \| issue = 12 \| year = 1961 \| pages = 1209–1225
	\| first = P. W. \| last = Kasteleyn		\| first = P. W. \| last = Kasteleyn
	\| author-link = ~~Pieter_Kasteleyn~~		\| author-link = Pieter Kasteleyn
	\| doi = 10.1016/0031-8914(61)90063-5\| bibcode = 1961Phy....27.1209K}}</ref> and Neville Temperley and Michael Fisher<ref>{{cite journal \|last1=Temperley \|first1=H. N. V. \|author-link1=Harold Neville Vazeille Temperley \|last2=Fisher \|first2=Michael E. \|author-link2=Michael Fisher \|year=1961 \|title=Dimer problem in statistical mechanics-an exact result \|journal=Philosophical Magazine \|volume=6 \|issue=68 \|pages=1061–1063 \|doi=10.1080/14786436108243366\|bibcode=1961PMag....6.1061T }}</ref> independently found the number of [[Domino tiling#Counting tilings of regions\|domino tilings]] for the ''m''-by-''n'' rectangle. This is equivalent to counting the number of perfect matchings for the ''m''-by-''n'' [[lattice graph]]. By 1967, Kasteleyn had generalized this result to all planar graphs.<ref>{{cite journal		\| doi = 10.1016/0031-8914(61)90063-5\| bibcode = 1961Phy....27.1209K}}</ref> and Neville Temperley and Michael Fisher<ref>{{cite journal \|last1=Temperley \|first1=H. N. V. \|author-link1=Harold Neville Vazeille Temperley \|last2=Fisher \|first2=Michael E. \|author-link2=Michael Fisher \|year=1961 \|title=Dimer problem in statistical mechanics-an exact result \|journal=Philosophical Magazine \|volume=6 \|issue=68 \|pages=1061–1063 \|doi=10.1080/14786436108243366\|bibcode=1961PMag....6.1061T }}</ref> independently found the number of [[Domino tiling#Counting tilings of regions\|domino tilings]] for the ''m''-by-''n'' rectangle. This is equivalent to counting the number of perfect matchings for the ''m''-by-''n'' [[lattice graph]]. By 1967, Kasteleyn had generalized this result to all planar graphs.<ref>{{cite journal
	\| last = Kasteleyn		\| last = Kasteleyn
	\| first = P. W.		\| first = P. W.
	\| author-link = ~~Pieter_Kasteleyn~~		\| author-link = Pieter Kasteleyn
	\| year = 1963		\| year = 1963
	\| title = Dimer Statistics and Phase Transitions		\| title = Dimer Statistics and Phase Transitions
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	}}</ref><ref>{{citation		}}</ref><ref>{{citation
	\| last = Kasteleyn \| first = P. W.		\| last = Kasteleyn \| first = P. W.
	\| author-link = ~~Pieter_Kasteleyn~~		\| author-link = Pieter Kasteleyn
	\| contribution = Graph theory and crystal physics		\| contribution = Graph theory and crystal physics
	\| editor-last = Harary \| editor-first = F. \| editor-link = Frank Harary		\| editor-last = Harary \| editor-first = F. \| editor-link = Frank Harary

InternetArchiveBot: Rescuing 1 sources and tagging 0 as dead.) #IABot (v2.0.9.5

2024-03-28T01:12:08Z

Rescuing 1 sources and tagging 0 as dead.) #IABot (v2.0.9.5

← Previous revision		Revision as of 01:12, 28 March 2024
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	==Applications==		==Applications==
	The FKT algorithm has seen extensive use in [[holographic algorithm]]s on planar graphs via [[matchgates]].<ref name=CLX/> For example, consider the planar version of the ice model mentioned above, which has the technical name #[[planar graph\|PL]]-3-NAE-[[Boolean satisfiability problem\|SAT]] (where NAE stands for "not all equal"). Valiant found a polynomial time algorithm for this problem which uses matchgates.<ref>{{cite conference		The FKT algorithm has seen extensive use in [[holographic algorithm]]s on planar graphs via [[matchgates]].<ref name=CLX/> For example, consider the planar version of the ice model mentioned above, which has the technical name #[[planar graph\|PL]]-3-NAE-[[Boolean satisfiability problem\|SAT]] (where NAE stands for "not all equal"). Valiant found a polynomial time algorithm for this problem which uses matchgates.<ref>{{cite conference
	\| title = Holographic Algorithms (Extended Abstract)		\|title = Holographic Algorithms (Extended Abstract)
	\| first = Leslie G.		\|first = Leslie G.
	\| last = Valiant		\|last = Valiant
	\| author-link = Leslie G. Valiant		\|author-link = Leslie G. Valiant
	\| year = 2004		\|year = 2004
	\| conference = FOCS'04		\|conference = FOCS'04
	\| conference-url = http://www.cs.brown.edu/~aris/focs04/		\|conference-url = http://www.cs.brown.edu/~aris/focs04/
	\| book-title = Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science		\|book-title = Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science
	\| publisher = IEEE Computer Society		\|publisher = IEEE Computer Society
	\| location = Rome, Italy		\|location = Rome, Italy
	\| pages = 306–315		\|pages = 306–315
	\| isbn = 0-7695-2228-9		\|isbn = 0-7695-2228-9
	\| doi = 10.1109/FOCS.2004.34		\|doi = 10.1109/FOCS.2004.34
			\|access-date = 2010-12-22
			\|archive-date = 2012-03-13
			\|archive-url = https://web.archive.org/web/20120313170241/http://www.cs.brown.edu/~aris/focs04/
			\|url-status = dead
	}}</ref>		}}</ref>

OAbot: Open access bot: hdl updated in citation with #oabot.

2024-01-03T22:58:54Z

Open access bot: hdl updated in citation with #oabot.

← Previous revision		Revision as of 22:58, 3 January 2024
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	\| issn = 0890-5401		\| issn = 0890-5401
	\| doi-access= free		\| doi-access= free
			\| hdl= 1813/6700
			\| hdl-access= free
	}}</ref> Since counting the number of perfect matchings in a general graph is [[Sharp-P-complete\|#P-complete]], some restriction on the input graph is required unless [[FP (complexity)\|FP]], the function version of [[P (complexity)\|P]], is equal to [[Sharp-P\|#P]]. Counting matchings, which is known as the [[Hosoya index]], is also #P-complete even for planar graphs.<ref>{{citation		}}</ref> Since counting the number of perfect matchings in a general graph is [[Sharp-P-complete\|#P-complete]], some restriction on the input graph is required unless [[FP (complexity)\|FP]], the function version of [[P (complexity)\|P]], is equal to [[Sharp-P\|#P]]. Counting matchings, which is known as the [[Hosoya index]], is also #P-complete even for planar graphs.<ref>{{citation
	\| last = Jerrum \| first = Mark \| author-link = Mark Jerrum		\| last = Jerrum \| first = Mark \| author-link = Mark Jerrum

ReyHahn: first names!

2023-10-11T23:42:38Z

first names!

← Previous revision		Revision as of 23:42, 11 October 2023
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	{{short description\|Algorithm for counting perfect matchings in planar graphs}}		{{short description\|Algorithm for counting perfect matchings in planar graphs}}
	The '''FKT algorithm''', named after [[Michael ~~Fisher\|~~Fisher]], [[Pieter ~~Kasteleyn\|~~Kasteleyn]], and [[Harold Neville Vazeille Temperley\|Temperley]], counts the number of [[perfect matching]]s in a [[planar graph\|planar]] graph in polynomial time. This same task is [[Sharp-P-complete\|#P-complete]] for general graphs. For [[Matching (graph theory)\|matching]]s that are not required to be perfect, counting them remains #P-complete even for planar graphs. The key idea of the FKT algorithm is to convert the problem into a [[Pfaffian]] computation of a [[skew-symmetric matrix]] derived from a planar embedding of the graph. The Pfaffian of this matrix is then computed efficiently using standard [[Determinant#Algorithmic implementation\|determinant algorithms]].		The '''Fisher–Kasteleyn–Temperley''' ('''FKT''') '''algorithm''', named after [[Michael Fisher]], [[Pieter Kasteleyn]], and [[Harold Neville Vazeille Temperley\|Neville Temperley]], counts the number of [[perfect matching]]s in a [[planar graph\|planar]] graph in polynomial time. This same task is [[Sharp-P-complete\|#P-complete]] for general graphs. For [[Matching (graph theory)\|matching]]s that are not required to be perfect, counting them remains #P-complete even for planar graphs. The key idea of the FKT algorithm is to convert the problem into a [[Pfaffian]] computation of a [[skew-symmetric matrix]] derived from a planar embedding of the graph. The Pfaffian of this matrix is then computed efficiently using standard [[Determinant#Algorithmic implementation\|determinant algorithms]].

	==History==		==History==
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	Another type of physical system to consider is composed of [[Dimer (chemistry)\|dimers]], which is a polymer with two atoms. The dimer model counts the number of dimer coverings of a graph.<ref>{{cite journal \|last1=Kenyon \|first1=Richard \|last2=Okounkov \|first2=Andrei \|year=2005 \|title=What is a Dimer? \|journal=AMS \|volume=52 \|issue=3 \|pages=342–343 \|url=https://www.ams.org/notices/200503/what-is.pdf}}</ref> Another physical system to consider is the bonding of [[H2O\|H<sub>2</sub>O]] molecules in the form of ice. This can be modelled as a directed, 3-[[regular graph\|regular]] graph where the orientation of the edges at each vertex cannot all be the same. How many edge orientations does this model have?		Another type of physical system to consider is composed of [[Dimer (chemistry)\|dimers]], which is a polymer with two atoms. The dimer model counts the number of dimer coverings of a graph.<ref>{{cite journal \|last1=Kenyon \|first1=Richard \|last2=Okounkov \|first2=Andrei \|year=2005 \|title=What is a Dimer? \|journal=AMS \|volume=52 \|issue=3 \|pages=342–343 \|url=https://www.ams.org/notices/200503/what-is.pdf}}</ref> Another physical system to consider is the bonding of [[H2O\|H<sub>2</sub>O]] molecules in the form of ice. This can be modelled as a directed, 3-[[regular graph\|regular]] graph where the orientation of the edges at each vertex cannot all be the same. How many edge orientations does this model have?

	Motivated by physical systems involving dimers, in 1961, Kasteleyn<ref>{{citation		Motivated by physical systems involving dimers, in 1961, Pieter Kasteleyn<ref>{{citation
	\| title = The statistics of dimers on a lattice. I. The number of dimer arrangements on a quadratic lattice		\| title = The statistics of dimers on a lattice. I. The number of dimer arrangements on a quadratic lattice
	\| journal = Physica		\| journal = Physica
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	\| first = P. W. \| last = Kasteleyn		\| first = P. W. \| last = Kasteleyn
	\| author-link = Pieter_Kasteleyn		\| author-link = Pieter_Kasteleyn
	\| doi = 10.1016/0031-8914(61)90063-5\| bibcode = 1961Phy....27.1209K}}</ref> and Temperley and Fisher<ref>{{cite journal \|last1=Temperley \|first1=H. N. V. \|author-link1=Harold Neville Vazeille Temperley \|last2=Fisher \|first2=Michael E. \|author-link2=Michael Fisher \|year=1961 \|title=Dimer problem in statistical mechanics-an exact result \|journal=Philosophical Magazine \|volume=6 \|issue=68 \|pages=1061–1063 \|doi=10.1080/14786436108243366\|bibcode=1961PMag....6.1061T }}</ref> independently found the number of [[Domino tiling#Counting tilings of regions\|domino tilings]] for the ''m''-by-''n'' rectangle. This is equivalent to counting the number of perfect matchings for the ''m''-by-''n'' [[lattice graph]]. By 1967, Kasteleyn had generalized this result to all planar graphs.<ref>{{cite journal		\| doi = 10.1016/0031-8914(61)90063-5\| bibcode = 1961Phy....27.1209K}}</ref> and Neville Temperley and Michael Fisher<ref>{{cite journal \|last1=Temperley \|first1=H. N. V. \|author-link1=Harold Neville Vazeille Temperley \|last2=Fisher \|first2=Michael E. \|author-link2=Michael Fisher \|year=1961 \|title=Dimer problem in statistical mechanics-an exact result \|journal=Philosophical Magazine \|volume=6 \|issue=68 \|pages=1061–1063 \|doi=10.1080/14786436108243366\|bibcode=1961PMag....6.1061T }}</ref> independently found the number of [[Domino tiling#Counting tilings of regions\|domino tilings]] for the ''m''-by-''n'' rectangle. This is equivalent to counting the number of perfect matchings for the ''m''-by-''n'' [[lattice graph]]. By 1967, Kasteleyn had generalized this result to all planar graphs.<ref>{{cite journal
	\| last = Kasteleyn		\| last = Kasteleyn
	\| first = P. W.		\| first = P. W.
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	Finally, for any [[skew-symmetric matrix]] ''A'',		Finally, for any [[skew-symmetric matrix]] ''A'',
	:<math>\operatorname{pf}(A)^2 = \det(A),</math>		:<math>\operatorname{pf}(A)^2 = \det(A),</math>
	where det(''A'') is the [[determinant]] of ''A''. This result is due to [[Arthur ~~Cayley\|~~Cayley]].<ref>{{cite journal		where det(''A'') is the [[determinant]] of ''A''. This result is due to [[Arthur Cayley]].<ref>{{cite journal
	\| last1 = Cayley		\| last1 = Cayley
	\| first1 = Arthur		\| first1 = Arthur

← Previous revision		Revision as of 23:29, 23 August 2024
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	{{short description\|Algorithm for counting perfect matchings in planar graphs}}		{{short description\|Algorithm for counting perfect matchings in planar graphs}}
			{{CS1 config\|mode=cs1}}
	The '''Fisher–Kasteleyn–Temperley''' ('''FKT''') '''algorithm''', named after [[Michael Fisher]], [[Pieter Kasteleyn]], and [[Harold Neville Vazeille Temperley\|Neville Temperley]], counts the number of [[perfect matching]]s in a [[planar graph\|planar]] graph in polynomial time. This same task is [[Sharp-P-complete\|#P-complete]] for general graphs. For [[Matching (graph theory)\|matching]]s that are not required to be perfect, counting them remains #P-complete even for planar graphs. The key idea of the FKT algorithm is to convert the problem into a [[Pfaffian]] computation of a [[skew-symmetric matrix]] derived from a planar embedding of the graph. The Pfaffian of this matrix is then computed efficiently using standard [[Determinant#Algorithmic implementation\|determinant algorithms]].		The '''Fisher–Kasteleyn–Temperley''' ('''FKT''') '''algorithm''', named after [[Michael Fisher]], [[Pieter Kasteleyn]], and [[Harold Neville Vazeille Temperley\|Neville Temperley]], counts the number of [[perfect matching]]s in a [[planar graph\|planar]] graph in polynomial time. This same task is [[Sharp-P-complete\|#P-complete]] for general graphs. For [[Matching (graph theory)\|matching]]s that are not required to be perfect, counting them remains #P-complete even for planar graphs. The key idea of the FKT algorithm is to convert the problem into a [[Pfaffian]] computation of a [[skew-symmetric matrix]] derived from a planar embedding of the graph. The Pfaffian of this matrix is then computed efficiently using standard [[Determinant#Algorithmic implementation\|determinant algorithms]].