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In the [[mathematics|mathematical]] discipline of [[graph theory]], a '''feedback vertex set''' of a [[graph (mathematics)|graph]] is a set of vertices whose removal leaves a graph without [[cycle (graph theory)|cycles]]. In other words, each feedback vertex set contains at least one vertex of any cycle in the graph.
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The '''feedback vertex set problem''' is an [[NP-complete]] problem in [[computational complexity theory]]. It was among the [[Karp's 21 NP-complete problems|first problems shown to be NP-complete]]. It has wide applications in [[operating system]]s, [[database system]]s, and [[VLSI]] chip design.


== Definition ==
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The [[decision problem]] is as follows:
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:INSTANCE: An (undirected or directed) [[Graph (mathematics)|graph]] <math>G = (V, E)</math> and a positive integer <math>k</math>.
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:QUESTION: Is there a subset <math>X \subseteq V</math> with <math>|X| \leq k</math> such that <math>G</math> with the vertices from <math>X</math> deleted is [[cycle (graph theory)|cycle-free]]?


The graph <math>G[V \setminus X]</math> that remains after removing <math>X</math> from <math>G</math> is an induced [[forest (graph theory)|forest]] (resp. an induced [[directed acyclic graph]] in the case of [[directed graph]]s). Thus, finding a minimum feedback vertex set in a graph is equivalent to finding a maximum induced forest (resp. maximum induced directed acyclic graph in the case of [[directed graph]]s).
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==NP-hardness==
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{{harvtxt|Karp|1972}} showed that the feedback vertex set problem for [[directed graph]]s is [[NP-complete]]. The problem remains NP-complete on directed graphs with maximum in-degree and out-degree two, and on directed planar graphs with maximum in-degree and out-degree three.<ref>unpublished results due to Garey and Johnson, cf. {{harvtxt|Garey|Johnson|1979}}: GT7</ref> Karp's reduction also implies the NP-completeness of the feedback vertex set problem on undirected graphs, where the problem stays NP-hard on graphs of [[Degree (graph theory)|maximum degree]] four. The feedback vertex set problem can be solved in polynomial time on graphs of [[Degree (graph theory)|maximum degree]] at most three.<ref>{{harvtxt|Ueno|Kajitani|Gotoh|1988}}; {{harvtxt|Li|Liu|1999}}</ref>
 
Note that the problem of deleting as few ''edges'' as possible to make the graph cycle-free is equivalent to finding a [[spanning tree]], which can be done in [[polynomial time]]. In contrast, the problem of deleting edges from a [[directed graph]] to make it [[Directed acyclic graph|acyclic]], the [[feedback arc set]] problem, is NP-complete.<ref name="k72">{{harvtxt|Karp|1972}}</ref>
 
==Exact algorithms==
The corresponding [[NP optimization problem]] of finding the size of a minimum feedback vertex set can be solved in time  ''O''(1.7347<sup>''n''</sup>), where  ''n'' is the number of vertices in the graph.<ref>{{harvtxt|Fomin|Villanger|2010}}</ref> This algorithm actually computes a maximum induced forest, and when such a forest is obtained, its complement is a minimum feedback vertex set.  The number of minimal feedback vertex sets in a graph is bounded by ''O''(1.8638<sup>''n''</sup>).{{sfnp|Fomin|Gaspers|Pyatkin|Razgon|2008}} The directed feedback vertex set problem can still be solved in time ''O*''(1.9977<sup>''n''</sup>), where&nbsp;''n'' is the number of vertices in the given directed graph.{{sfnp|Razgon|2007}} The parameterized versions of the directed and undirected problems are both [[fixed-parameter tractable]].{{sfnp|Chen|Liu|Lu|O'Sullivan|2008}}
 
==Approximation==
The problem is [[APX|APX-complete]], which directly follows from the APX-completeness of the [[vertex cover|vertex cover problem]],<ref>{{harvnb|Dinur|Safra|2005}}</ref> and the existence of an approximation preserving [[L-reduction]] from the vertex cover problem to it.<ref name="k72"/> The best known approximation algorithm on undirected graphs is by a factor of two, based on the Becker-Geiger theorem.{{sfnp|Becker|Geiger|1996}}
 
== Bounds ==
According to the [[Erdős–Pósa theorem]], the size of a minimum feedback vertex set is within a logarithmic factor of the maximum number of vertex-disjoint cycles in the given graph.{{sfnp|Erdős|Pósa|1965}}
 
== Applications ==
In [[operating system]]s, feedback vertex sets play a prominent role in the study of [[deadlock]] recovery. In the [[wait-for graph]] of an operating system, each directed cycle corresponds to a deadlock situation. In order to resolve all deadlocks, some blocked processes have to be aborted. A minimum feedback vertex set in this graph corresponds to a minimum number of processes that one needs to abort.{{sfnp|Silberschatz|Galvin|Gagne|2008}}
 
Furthermore, the feedback vertex set problem has applications in [[VLSI]] chip design.{{sfnp|Festa|Pardalos|Resende|2000}}
 
==Notes==
{{reflist|colwidth=25em}}
 
== References ==
 
===Research articles===
{{refbegin|colwidth=25em}}
*{{citation
| last1 = Becker | first1 = Ann
| last2 = Bar-Yehuda | first2 = Reuven
| last3 = Geiger | first3 = Dan
| arxiv = 1106.0225
| doi = 10.1613/jair.638
| journal = [[Journal of Artificial Intelligence Research]]
| mr = 1765590
| pages = 219–234
| title = Randomized algorithms for the loop cutset problem
| volume = 12
| year = 2000}}
*{{citation
| last1 = Becker | first1 = Ann
| last2 = Geiger | first2 = Dan
| title = Optimization of Pearl's method of conditioning and greedy-like approximation algorithms for the vertex feedback set problem.
| journal = [[Artificial Intelligence (journal)|Artificial Intelligence]]
| volume = 83
| issue = 1
| year = 1996
| pages = 167–188
| doi = 10.1016/0004-3702(95)00004-6 }}
*{{citation
| last1 = Cao | first1 = Yixin
| last2 = Chen | first2 = Jianer
| last3 = Liu | first3 = Yang
| editor-last = Kaplan | editor-first = Haim
| contribution = On feedback vertex set: new measure and new structures
| doi = 10.1007/978-3-642-13731-0_10
| pages = 93–104
| series = Lecture Notes in Computer Science
| title = Proc. 12th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2010), Bergen, Norway, June 21-23, 2010
| volume = 6139
| year = 2010}}
*{{citation
| last1 = Chen | first1 = Jianer
| last2 = Fomin | first2 = Fedor V.
| last3 = Liu | first3 = Yang
| last4 = Lu | first4 = Songjian
| last5 = Villanger | first5 = Yngve
| doi = 10.1016/j.jcss.2008.05.002
| issue = 7
| journal = [[Journal of Computer and System Sciences]]
| mr = 2454063
| pages = 1188–1198
| title = Improved algorithms for feedback vertex set problems
| volume = 74
| year = 2008}}
*{{citation
| last1 = Chen | first1 = Jianer
| last2 = Liu | first2 = Yang
| last3 = Lu | first3 = Songjian
| last4 = O'Sullivan | first4 = Barry
| last5 = Razgon | first5 = Igor
| doi = 10.1145/1411509.1411511
| issue = 5
| journal = [[Journal of the ACM]]
| mr = 2456546
| at = Art. 21
| title = A fixed-parameter algorithm for the directed feedback vertex set problem
| volume = 55
| year = 2008}}
*{{citation
| last1 = Dinur | first1 = Irit | author1-link = Irit Dinur
| last2 = Safra | first2 = Samuel
| doi = 10.4007/annals.2005.162.439
| issue = 1
| journal = [[Annals of Mathematics]]
| mr = 2178966
| pages = 439–485
| series = Second Series
| title = On the hardness of approximating minimum vertex cover
| url = http://www.cs.huji.ac.il/~dinuri/mypapers/vc.pdf
| volume = 162
| year = 2005}}
*{{citation
| last1 = Erdős | first1 = Paul | author1-link = Paul Erdős
| last2 = Pósa | first2 = Lajos | author2-link = Lajos Pósa (mathematician)
| doi = 10.4153/CJM-1965-035-8
| journal = [[Canadian Journal of Mathematics]]
| pages = 347–352
| title = On independent circuits contained in a graph
| url = http://www.renyi.hu/~p_erdos/1965-05.pdf
| volume = 17
| year = 1965}}
*{{citation
| last1 = Fomin | first1 = Fedor V.
| last2 = Gaspers | first2 = Serge
| last3 = Pyatkin | first3 = Artem
| last4 = Razgon | first4 = Igor
| title = On the minimum feedback vertex set problem: exact and enumeration algorithms.
| journal = [[Algorithmica]]
| volume = 52
| issue = 2
| year = 2008
| pages = 293–307
| doi = 10.1007/s00453-007-9152-0}}
*{{citation
| last1 = Fomin | first1 = Fedor V.
| last2 = Villanger | first2 = Yngve
| contribution = Finding induced subgraphs via minimal triangulations
| series = Leibniz International Proceedings in Informatics (LIPIcs)
| title = Proc. 27th International Symposium on Theoretical Aspects of Computer Science (STACS 2010)
| volume = 5
| year = 2010
| pages = 383–394
| doi = 10.4230/LIPIcs.STACS.2010.2470}}
*{{citation
| last = Karp | first = Richard M. | authorlink = Richard Karp
| contribution = Reducibility Among Combinatorial Problems
| location = New York
| pages = 85–103
| publisher = Plenum
| title = Proc. Symposium on Complexity of Computer Computations, IBM Thomas J. Watson Res. Center, Yorktown Heights, N.Y.
| year = 1972}}
*{{citation
| last1 = Li | first1 = Deming
| last2 = Liu | first2 = Yanpei
| issue = 4
| journal = Acta Mathematica Scientia
| mr = 1735603
| pages = 375–381
| title = A polynomial algorithm for finding the minimum feedback vertex set of a 3-regular simple graph
| volume = 19
| year = 1999}}
*{{citation
| last = Razgon | first = I.
| editor1-last = Italiano | editor1-first = Giuseppe F.
| editor2-last = Moggi | editor2-first = Eugenio
| editor3-last = Laura | editor3-first = Luigi
| contribution = Computing minimum directed feedback vertex set in O*(1.9977<sup>''n''</sup>)
| pages = 70–81
| publisher = World Scientific
| title = Proceedings of the 10th Italian Conference on Theoretical Computer Science
| url = http://www.cs.le.ac.uk/people/ir45/papers/ictcsIgorCamera.pdf
| year = 2007}}
*{{citation
| last1 = Ueno | first1 = Shuichi
| last2 = Kajitani | first2 = Yoji
| last3 = Gotoh | first3 = Shin'ya
| doi = 10.1016/0012-365X(88)90226-9
| issue = 1-3
| journal = [[Discrete Mathematics (journal)|Discrete Mathematics]]
| mr = 975556
| pages = 355–360
| title = On the nonseparating independent set problem and feedback set problem for graphs with no vertex degree exceeding three
| volume = 72
| year = 1988}}
{{refend}}
 
===Textbooks and survey articles===
*{{citation
| last1 = Festa | first1 = P.
| last2 = Pardalos | first2 = P. M.
| last3 = Resende | first3 = M.G.C.
| editor1-last = Du | editor1-first = D.-Z.
| editor2-last = Pardalos | editor2-first = P. M.
| contribution = Feedback set problems
| pages = 209–259
| publisher = Kluwer Academic Publishers
| title = Handbook of Combinatorial Optimization, Supplement vol. A
| url = http://www.research.att.com/~mgcr/doc/sfsp.pdf
| year = 2000}}
*{{citation
| last1 = Garey | first1 = Michael R. | author1-link = Michael R. Garey
| last2 = Johnson | first2 = David S. | author2-link = David S. Johnson
| at =  A1.1: GT7, p.&nbsp;191
| isbn = 0-7167-1045-5
| publisher = W.H. Freeman
| title = [[Computers and Intractability: A Guide to the Theory of NP-Completeness]]
| year = 1979}}
*{{citation
| last1 = Silberschatz | first1 = Abraham
| last2 = Galvin | first2 = Peter Baer
| last3 = Gagne | first3 = Greg
| edition = 8th
| isbn = 978-0-470-12872-5
| publisher = John Wiley & Sons. Inc
| title = Operating System Concepts
| year = 2008}}
 
{{DEFAULTSORT:Feedback Vertex Set}}
[[Category:NP-complete problems]]
[[Category:Computational problems in graph theory]]

Latest revision as of 00:28, 12 November 2014

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