feedback vertex set

The FVS decision problem is as follows:

:INSTANCE: An (undirected or directed) graph $G\; =\; (V,\; E)$ and a positive integer $k$.

:QUESTION: Is there a subset $X\; \backslash subseteq\; V$ with $|X|\; \backslash leq\; k$ such that, when all vertices of $X$ and their adjacent edges are deleted from $G$, the remainder is cycle-free?

The graph $G[V\; \backslash setminus\; X]$ that remains after removing $X$ from $G$ is an induced forest (resp. an induced directed acyclic graph in the case of directed graphs). Thus, finding a minimum FVS in a graph is equivalent to finding a maximum induced forest (resp. maximum induced directed acyclic graph in the case of directed graphs).

{{harvtxt|Karp|1972}} showed that finding a feedback vertex set of size $\backslash le\; k$ in directed graphs 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.unpublished results due to Garey and Johnson, cf. {{harvtxt|Garey|Johnson|1979}}: GT7

Karp's reduction also implies the NP-completeness of the feedback vertex set problem on undirected graphs, where the problem stays NP-complete on graphs of maximum degree four. The feedback vertex set problem can be solved in polynomial time on graphs of maximum degree at most three, using an algorithm based on the matroid parity problem.{{harvtxt|Ueno|Kajitani|Gotoh|1988}}; {{harvtxt|Li|Liu|1999}}

The corresponding NP optimization problem of finding the size of a minimum feedback vertex set can be solved in time O(1.7347ⁿ), where n is the number of vertices in the graph.{{harvtxt|Fomin|Villanger|2010}} 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(1.8638ⁿ).{{sfnp|Fomin|Gaspers|Pyatkin|Razgon|2008}} The directed feedback vertex set problem can still be solved in time O*(1.9977ⁿ), where 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}}

In undirected graphs of maximum degree three, the feedback vertex set problem can be solved in polynomial time, by transforming it into an instance of the matroid parity problem for linear matroids.{{sfnp|Ueno|Kajitani|Gotoh|1988}}

The undirected problem is APX-complete. This follows from the following facts.

• The APX-completeness of the vertex cover problem;{{harvnb|Dinur|Safra|2005}}
• The existence of an approximation preserving L-reduction from the vertex cover problem to it;
• Existing constant-factor approximation algorithms.{{harvtxt|Karp|1972}}

The best known approximation algorithm on undirected graphs is by a factor of two.{{harvtxt|Becker|Geiger|1996}}. See also {{harvtxt|Bafna|Berman|Fujito|1999}} for an alternative approximation algorithm with the same approximation ratio.

By contrast, the directed version of the problem appears to be much harder to approximate. Under the unique games conjecture, an unproven but commonly used computational hardness assumption, it is NP-hard to approximate the problem to within any constant factor in polynomial time. The same hardness result was originally proven for the closely related feedback arc set problem,{{Cite book |last1=Guruswami |first1=Venkatesan |last2=Manokaran |first2=Rajsekar |last3=Raghavendra |first3=Prasad |title=2008 49th Annual IEEE Symposium on Foundations of Computer Science |chapter=Beating the Random Ordering is Hard: Inapproximability of Maximum Acyclic Subgraph |date=2008 |chapter-url=https://ieeexplore.ieee.org/document/4690990 |pages=573–582 |doi=10.1109/FOCS.2008.51|isbn=978-0-7695-3436-7 |s2cid=8762205 }} but since the feedback arc set problem and feedback vertex set problem in directed graphs are reducible to one another while preserving solution sizes,{{Cite journal |last1=Even |first1=G. |last2=(Seffi) Naor |first2=J. |last3=Schieber |first3=B. |last4=Sudan |first4=M. |date=1998 |title=Approximating Minimum Feedback Sets and Multicuts in Directed Graphs |url=http://link.springer.com/10.1007/PL00009191 |journal=Algorithmica |language=en |volume=20 |issue=2 |pages=151–174 |doi=10.1007/PL00009191 |s2cid=2437790 |issn=0178-4617}} it also holds for the latter.

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}}

• Instead of vertices, one can consider a feedback edge set - a set of edges in an undirected graph, whose removal makes the graph acyclic. The size of a smallest feedback edge set in a graph is called the circuit rank of the graph. In contrast to the FVS number, the circuit rank can be easily computed: it is $|E|-|V|+|C|$, where C is the set of connected components of the graph. The problem of finding a smallest feedback edge set is equivalent to finding a spanning forest, which can be done in polynomial time.
• The analogous concept in a directed graph is the feedback arc set (FAS) - a set of directed arcs whose removal makes the graph acyclic. Finding a smallest FAS is an NP-hard problem.

• In operating systems, 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}}
• The feedback vertex set problem has applications in VLSI chip design.{{sfnp|Festa|Pardalos|Resende|2000}}
• Another application is in complexity theory. Some computational problems on graphs are NP-hard in general, but can be solved in polynomial time for graphs with bounded FVS number. Some examples are graph isomorphism{{Cite book|last1=Kratsch|first1=Stefan|last2=Schweitzer|first2=Pascal|title=Algorithm Theory - SWAT 2010 |chapter=Isomorphism for Graphs of Bounded Feedback Vertex Set Number |date=2010|editor-last=Kaplan|editor-first=Haim|chapter-url=https://link.springer.com/chapter/10.1007/978-3-642-13731-0_9|series=Lecture Notes in Computer Science|volume=6139 |language=en|location=Berlin, Heidelberg|publisher=Springer|pages=81–92|doi=10.1007/978-3-642-13731-0_9|bibcode=2010LNCS.6139...81K |isbn=978-3-642-13731-0}} and the path reconfiguration problem.{{Cite book| series=Lecture Notes in Computer Science | year=2019 | volume=11646 |url=https://link.springer.com/content/pdf/10.1007/978-3-030-24766-9.pdf#page=370|language=en-gb|doi=10.1007/978-3-030-24766-9| isbn=978-3-030-24765-2 | s2cid=198996919 | title=Algorithms and Data Structures }}

{{reflist|colwidth=25em}}

{{refbegin|colwidth=25em}}

• {{citation

| last1 = Bafna | first1 = Vineet

| last2 = Berman | first2 = Piotr

| last3 = Fujito | first3 = Toshihiro

| doi = 10.1137/S0895480196305124

| issue = 3

| journal = SIAM Journal on Discrete Mathematics

| mr = 1710236

| pages = 289–297 (electronic)

| title = A 2-approximation algorithm for the undirected feedback vertex set problem

| volume = 12

| year = 1999

}}.

• {{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| s2cid = 10243677

}}

• {{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

| volume = 83

| issue = 1

| year = 1996

| pages = 167–188

| doi = 10.1016/0004-3702(95)00004-6 | citeseerx = 10.1.1.25.1442

}}

• {{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| arxiv = 1004.1672| bibcode = 2010LNCS.6139...93C| isbn = 978-3-642-13730-3

}}

• {{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| s2cid = 1547510

}}

• {{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

| access-date = 2010-03-29

| archive-date = 2009-09-20

| archive-url = https://web.archive.org/web/20090920071048/http://www.cs.huji.ac.il/~dinuri/mypapers/vc.pdf

| url-status = dead

}}

• {{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

| s2cid = 123981328

}}

• {{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| citeseerx = 10.1.1.722.8913

| s2cid = 731997

}}

• {{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| doi-access = free

| isbn = 9783939897163

| s2cid = 436224

}}

• {{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

| doi=10.1016/s0252-9602(17)30520-9}}

• {{citation

| last = Razgon

| first = I.

| editor1-last = Italiano

| editor1-first = Giuseppe F.

| editor1-link = Giuseppe F. Italiano

| editor2-last = Moggi

| editor2-first = Eugenio

| editor3-last = Laura

| editor3-first = Luigi

| contribution = Computing minimum directed feedback vertex set in O*(1.9977ⁿ)

| 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

| 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| doi-access = free

}}

{{refend}}

• {{citation

| last1 = Festa | first1 = P.

| last2 = Pardalos | first2 = P. M.

| last3 = Resende | first3 = M.G.C. | author3-link = Mauricio Resende

| 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 = [https://archive.org/details/computersintract0000gare/page/ A1.1: GT7, p. 191] | isbn = 978-0-7167-1045-5 | publisher = W.H. Freeman | title = Computers and Intractability: A Guide to the Theory of NP-Completeness | year = 1979 | title-link = Computers and Intractability: A Guide to the Theory of NP-Completeness }}
• {{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