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| In [[graph theory]], a '''cut''' is a [[Partition_of_a_set|partition]] of the vertices of a graph into two disjoint subsets that are joined by at least one edge. The '''cut-set''' of the cut is the set of edges whose end points are in different subsets of the partition. Edges are said to be ''crossing'' the cut if they are in its cut-set.
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| (Many graph theorists use the term '''cut''' to refer to the edge set instead of the vertex partition.)
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| In an unweighted undirected graph, the ''size'' or ''weight'' of a cut is the number of edges crossing the cut. In a [[Graph_(mathematics)#Weighted_graph| weighted graph]], the same term is defined by the sum of the weights of the edges crossing the cut.
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| In a [[flow network]], an '''s-t cut''' is a cut that requires the [[Glossary_of_graph_theory#Direction|''source'']] and the [[Glossary_of_graph_theory#Direction|''sink'']] to be in different subsets, and its ''cut-set'' only consists of edges going from the source's side to the sink's side. The ''capacity'' of an s-t cut is defined as the sum of [[Capacity of a set|capacity]] of each edge in the ''cut-set''.
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| ==Definition==
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| : A '''cut''' <math>C=(S,T)</math> is a partition of <math>V</math> of a graph <math>G=(V,E)</math>. | |
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| : An '''s-t cut''' <math>C=(S,T)</math> of a network <math>N=(V,E)</math> is a cut of <math>N</math> such that <math>s\in S</math> and <math>t \in T</math>, where <math>s</math> and <math>t</math> are the [[Glossary_of_graph_theory#Direction|''source'']] and the [[Glossary_of_graph_theory#Direction|''sink'']] of <math>N</math> respectively. | |
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| : The '''cut-set''' of a cut <math>C=(S,T)</math> is the set <math>\{(u,v)\in E | u\in S, v \in T\}</math>. | |
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| :The '''size''' of a cut <math>C=(S,T)</math> is the number of edges in the cut-set. If the edges are weighted, the '''value''' (or '''weight''') of the cut is the sum of the weights. | |
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| ==Minimum cut==
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| [[File:Min-cut.svg|frame|right|A minimum cut.]]
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| {{main|Minimum cut}}
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| A cut is ''minimum'' if the size of the cut is not larger than the size of any other cut. The illustration on the right shows a minimum cut: the size of this cut is 2, and there is no cut of size 1 because the graph is [[Bridge (graph theory)|bridgeless]].
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| The [[max-flow min-cut theorem]] proves that the maximum [[flow network|network flow]] and the sum of the cut-edge weights of any minimum cut that separates the source and the sink are equal. There are [[polynomial time|polynomial-time]] methods to solve the min-cut problem, notably the [[Edmonds-Karp algorithm]].
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| {{clear}}
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| ==Maximum cut==
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| [[File:Max-cut.svg|frame|right|A maximum cut.]]
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| {{main|Maximum cut}}
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| A cut is ''maximum'' if the size of the cut is not smaller than the size of any other cut. The illustration on the right shows a maximum cut: the size of the cut is equal to 5, and there is no cut of size |''E''| because the graph is not [[Bipartite graph|bipartite]] (there is an [[Cycle graph#Terminology|odd cycle]]).
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| In general, finding a maximum cut is computationally hard. The max-cut problem is one of [[Karp's 21 NP-complete problems]]. The max cut problem is also [[Constant-factor approximation algorithm|APX-hard]], meaning that there is no polynomial-time approximation scheme for it unless P = NP.
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| Note that min-cut and max-cut are ''not'' [[Linear programming#Duality|dual]] problems in the [[linear programming]] sense, even though one gets from one problem to other by changing min to max in the [[objective function]]. The max-flow problem is the dual of the min-cut problem
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| == Sparsest cut ==
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| The '''Sparsest cut problem''' is to bipartition the vertices so as to minimize the ratio of the number of edges across the cut divided by the number of vertices in the smaller half of the partition. This objective function favors solutions that are both sparse (few edges crossing the cut) and balanced (close to a bisection). The problem is known to be NP-Hard, and the best known algorithm is an <math>O(\sqrt{\log n})</math> approximation due to {{Harvtxt|Arora|Rao|Vazirani|2009}}.
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| == See also ==
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| * [[Connectivity (graph theory)]]
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| * [[Graph cuts in computer vision]]
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| * [[Vertex separator]]
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| == References ==
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| *{{Citation | last1=Arora | first1=Sanjeev | author1-link=Sanjeev Arora | last2=Rao | first2=Satish | last3=Vazirani | first3=Umesh | author3-link=Umesh Vazirani | title=Expander flows, geometric embeddings and graph partitioning | publisher=ACM | year=2009 | journal=J. ACM | issn=0004-5411 | doi=10.1145/1502793.1502794 | volume=56 | issue=2 | pages=1–37}}
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| *{{cite book | author = [[Thomas H. Cormen]], [[Charles E. Leiserson]], [[Ronald L. Rivest]], and [[Clifford Stein]] | title=[[Introduction to Algorithms]], Second Edition | year = 2001 | publisher = MIT Press and McGraw-Hill | isbn = 0-262-03293-7 | pages=563,655,1043}}
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| *{{cite book|author = [[Michael R. Garey]] and [[David S. Johnson]] | year = 1979 | title = [[Computers and Intractability: A Guide to the Theory of NP-Completeness]] | publisher = W.H. Freeman | isbn = 0-7167-1045-5}} A2.2: ND16, pg.210.
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| *M. X. Goemans, and D. P. Williamson, [http://portal.acm.org/citation.cfm?id=227684 ''Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming''], Journal of the ACM, 42, 6 (Nov. 1995), 1115-1145
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| *[[Richard Karp|R. M. Karp]], ''Reducibility among combinatorial problems'', in R. E. Miller and J. W. Thacher (eds.), ''Complexity of Computer Computation'', Plenum Press, New York, 85-103 (1972)
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| *[[Subhash Khot|S. Khot]], G. Kindler, E. Mossel, and R. O’Donnell, [http://www.cs.cmu.edu/~odonnell/papers/maxcut.pdf ''Optimal inapproximability results for MAX-CUT and other two-variable CSPs?''], In Proceedings of the 45th IEEE Symposium on Foundations of Computer Science, pp. 146–154, 2004.
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| *{{cite book | author = [[Vijay_Vazirani | Vijay V. Vazirani]] | title = Approximation Algorithms | year = 2004 | publisher = Springer | isbn = 3-540-65367-8 | pages = 97–98}}
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| *{{cite web|last=Meira|first=Luis A. A.|title=Semidefinite Programming Based Algorithms for the Sparsest Cut Problem|url=http://www.ic.unicamp.br/~fkm/publication/Rairo11.pdf|accessdate=6 September 2011}}
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| [[Category:Graph connectivity]]
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| [[Category:Combinatorial optimization]]
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