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| In mathematics, the '''minimum ''k''-cut''', is a [[combinatorial optimization]] problem that requires finding a set of edges whose removal would partition the graph to ''k'' connected components. These edges are referred to as '''''k''-cut'''. The goal is to find the minimum-weight ''k''-cut. This partitioning can have applications in [[VLSI]] design, [[data-mining]], [[finite elements]] and communication in [[parallel computing]].
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| ==Formal definition==
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| Given an undirected graph ''G'' = (''V'', ''E'') with an assignment of weights to the edges ''w'': ''E'' → ''N'' and an integer ''k'' ∈ {2, 3, …, |''V''|}, partition ''V'' into ''k'' disjoint sets ''F'' = {''C''<sub>1</sub>, ''C''<sub>2</sub>, …, ''C''<sub>''k''</sub>} while minimizing
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| : <math>\sum_{i=1}^{k-1}\sum_{j=i+1}^k\sum_{\begin{smallmatrix} v_1 \in C_i \\ v_2 \in C_j \end{smallmatrix}} w ( \left \{ v_1, v_2 \right \} )</math>
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| For a fixed ''k'', the problem is [[polynomial time]] solvable in ''O''(|''V''|<sup>''k''<sup>2</sup></sup>).<ref>{{harvnb|Goldschmidt|Hochbaum|1988}}.</ref> However, the problem is [[NP-complete]] if ''k'' is part of the input.<ref>{{harvnb|Garey|Johnson|1979}}</ref> It is also NP-complete if we specify <math>k</math> vertices and ask for the minimum <math>k</math>-cut which separates these vertices among each of the sets.<ref>[http://www.jstor.org/stable/3690374?seq=13], which cites [http://citeseer.ist.psu.edu/viewdoc/summary?doi=10.1.1.13.6534]</ref>
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| ==Approximations==
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| Several [[approximation algorithms]] exist with an approximation of 2 − 2/''k''. A simple [[greedy algorithm]] that achieves this approximation factor computes a [[minimum cut]] in each connected components and removes the lightest one. This algorithm requires a total of ''n'' − 1 [[max flow]] computations. Another algorithm achieving the same guarantee uses the [[Gomory–Hu tree]] representation of minimum cuts. Constructing the Gomory–Hu tree requires ''n'' − 1 max flow computations, but the algorithm requires an overall ''O''(''kn'') max flow computations. Yet, it is easier to analyze the approximation factor of the second algorithm.<ref>{{harvnb|Saran|Vazirani|1991}}.</ref><ref>{{harvnb|Vazirani|2003|pp=40–44}}.</ref>
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| If we restrict the graph to a metric space, meaning a [[complete graph]] that satisfies the [[triangle inequality]], and enforce that the output partitions are each of pre-specified sizes, the problem is approximable to within a factor of 3 for any fixed ''k''.<ref>{{harvnb|Guttmann-Beck|Hassin|1999|pp=198–207}}.</ref>
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| ==See also==
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| * [[Maximum cut]]
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| * [[Minimum cut]]
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| ==Notes==
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| <references />
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| ==References==
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| * {{citation
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| | first1=O.
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| | last1=Goldschmidt
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| | first2=D. S.
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| | last2=Hochbaum
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| | title=Proc. 29th Ann. IEEE Symp. on Foundations of Comput. Sci.
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| | publisher=IEEE Computer Society
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| | year=1988
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| | pages=444–451 }}
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| * {{citation
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| | first1=M. R.
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| | last1=Garey
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| | first2=D. S.
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| | last2=Johnson
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| | title=Computers and Intractability: A Guide to the Theory of NP-Completeness
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| | publisher = W.H. Freeman
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| | year=1979
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| | isbn = 0-7167-1044-7 }}
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| * {{citation
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| | first1=H.
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| | last1=Saran
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| | first2=V.
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| | last2=Vazirani
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| | contribution=Finding ''k''-cuts within twice the optimal
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| | contribution-url=http://www.cc.gatech.edu/~vazirani/k-cut.ps
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| | title=Proc. 32nd Ann. IEEE Symp. on Foundations of Comput. Sci
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| | publisher=IEEE Computer Society
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| | year=1991
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| | pages=743–751 }}
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| * {{citation
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| | last = Vazirani
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| | first = Vijay V.
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| | authorlink = Vijay Vazirani
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| | title = Approximation Algorithms
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| | publisher = Springer
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| | year = 2003
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| | location = Berlin
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| | isbn = 3-540-65367-8 }}
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| * {{citation
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| | last1 = Guttmann-Beck
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| | first1 = N.
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| | last2 = Hassin
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| | first2 = R.
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| | contribution = Approximation algorithms for minimum ''k''-cut
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| | contribution-url = http://www.math.tau.ac.il/~hassin/k_cut_00.pdf
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| | title = Algorithmica
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| | year = 1999
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| | pages = 198–207 }}
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| * {{citation
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| | last1 = Comellas
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| | first1 = Francesc
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| | last2 = Sapena
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| | first2 = Emili
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| | title = A multiagent algorithm for graph partitioning. Lecture Notes in Comput. Sci.
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| | year = 2006
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| | volume = 3907
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| | pages = 279–285
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| | issn = 0302-9743
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| | doi= 10.1007/s004530010013
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| | journal = Algorithmica
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| | issue = 2
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| | url=http://www-ma4.upc.es/~comellas/evocomnet06/CoSa06-EvoCOMNET06.html}}
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| * {{citation
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| | last1=Crescenzi | first1=Pierluigi
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| | last2=Kann | first2=Viggo
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| | last3=Halldórsson | first3=Magnús
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| | last4=Karpinski | first4=Marek | authorlink4=Marek Karpinski
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| | last5=Woeginger | first5=Gerhard
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| | title=A Compendium of NP Optimization Problems
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| | contribution=Minimum k-cut
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| | url=http://www.csc.kth.se/~viggo/wwwcompendium/node90.html
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| | year=2000 }}
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| [[Category:NP-complete problems]]
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| [[Category:Combinatorial optimization]]
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| [[Category:Computational problems in graph theory]]
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| [[Category:Approximation algorithms]]
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