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| [[Image:Max-cut.svg|frame|right|A maximum cut.]]
| | If you present photography effectively, it helps you look much more properly at the globe around you. It is thus, on these grounds that compel various web service provider companies to integrate the same in their packages too. Your parishioners and certainly interested audience can come in to you for further information from the group and sometimes even approaching happenings and systems with the church. In the recent years, there has been a notable rise in the number of companies hiring Indian Word - Press developers. You can customize the appearance with PSD to Word - Press conversion ''. <br><br> |
| For a [[graph (mathematics)|graph]], a '''maximum cut''' is a [[cut (graph theory)|cut]] whose size is at least the size of any other cut. The problem of finding a maximum cut in a graph is known as the '''max-cut problem.'''
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| The problem can be stated simply as follows. One wants a subset ''S'' of the vertex set such that the number of edges between ''S'' and the complementary subset is as large as possible.
| | Thus, it is imperative that you must Hire Word - Press Developers who have the expertise and proficiency in delivering theme integration and customization services. While direct advertising is limited to few spots in your site and tied to fixed monthly payment by the advertisers, affiliate marketing can give you unlimited income as long as you can convert your traffic to sales. A Wordpress plugin is a software that you can install into your Wordpress site. You can add new functionalities and edit the existing ones to suit your changing business needs. For a Wordpress website, you don't need a powerful web hosting account to host your site. <br><br>But before choosing any one of these, let's compare between the two. s cutthroat competition prevailing in the online space won. We can active Akismet from wp-admin > Plugins > Installed Plugins. Every single Theme might be unique, providing several alternatives for webpage owners to reap the benefits of in an effort to instantaneously adjust their web page appear. If you loved this article and you also would like to collect more info relating to [http://www.busgamesforkids.com/groups/ wordpress dropbox backup] please visit our own website. " Thus working with a Word - Press powered web application, making any changes in the website design or website content is really easy and self explanatory. <br><br>You can add keywords but it is best to leave this alone. Russell HR Consulting provides expert knowledge in the practical application of employment law as well as providing employment law training and HR support services. Some examples of its additional features include; code inserter (for use with adding Google Analytics, Adsense section targeting etc) Webmaster verification assistant, Link Mask Generator, Robots. Fast Content Update - It's easy to edit or add posts with free Wordpress websites. Fortunately, Word - Press Customization Service is available these days, right from custom theme design, to plugin customization and modifying your website, you can take any bespoke service for your Word - Press development project. <br><br>More it extends numerous opportunities where your firm is at comfort and rest assured of no risks & errors. An ease of use which pertains to both internet site back-end and front-end users alike. However, you must also manually approve or reject comments so that your website does not promote parasitic behavior. Page speed is an important factor in ranking, especially with Google. For your information, it is an open source web content management system. |
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| There is a more advanced version of the problem called '''weighted max-cut'''. In this version each edge has a real number, its '''weight''', and the objective is to maximize not the number of edges but the total weight of the edges between ''S'' and its complement. The weighted max-cut problem is often, but not always, restricted to non-negative weights, because negative weights can change the nature of the problem.
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| ==Computational complexity==
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| The following [[decision problem]] related to maximum cuts has been studied widely in [[theoretical computer science]]:
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| :Given a graph ''G'' and an integer ''k'', determine whether there is a cut of size at least ''k'' in ''G''.
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| This problem is known to be [[NP-complete]]. It is easy to see that the problem is in [[NP (complexity)|NP]]: a ''yes'' answer is easy to prove by presenting a large enough cut. The NP-completeness of the problem can be shown, for example, by a transformation from [[maximum 2-satisfiability]] (a restriction of the [[maximum satisfiability problem]]).<ref>{{harvtxt|Garey|Johnson|1979}}.</ref> The weighted version of the decision problem was one of [[Karp's 21 NP-complete problems]];<ref>{{harvtxt|Karp|1972}}.</ref> Karp showed the NP-completeness by a reduction from the [[partition problem]].
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| The canonical [[optimization problem|optimization variant]] of the above decision problem is usually known as the ''maximum cut problem'' or ''max-cut problem'' and is defined as:
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| :Given a graph ''G'', find a maximum cut.
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| ==Polynomial-time algorithms==
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| As the max-cut problem is NP-hard, no polynomial-time algorithms for max-cut in general graphs are known.
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| However, in [[planar graph]]s, the maximum cut problem is dual to the [[route inspection problem]] (the problem of finding a shortest tour that visits each edge of a graph exactly once), in the sense that the edges that do not belong to a maximum cut of a graph ''G'' are the duals of the edges that are doubled in an optimal inspection tour of the [[dual graph]] of ''G''. The optimal inspection tour forms a self-intersecting curve that separates the plane into two subsets, the subset of points for which the [[winding number]] of the curve is even and the subset for which the winding number is odd; these two subsets form a cut that includes all of the edges whose duals appear an odd number of times in the tour. The route inspection problem may be solved in polynomial time, and this duality allows the maximum cut problem to also be solved in polynomial time for planar graphs.<ref>{{harvtxt|Hadlock|1975}}.</ref>
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| ==Approximation algorithms==
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| The max-cut problem is [[Constant-factor approximation algorithm|APX-hard]],<ref>{{harvtxt|Papadimitriou|Yannakakis|1991}} prove [[MaxSNP]]-completeness.</ref> meaning that there is no polynomial-time approximation scheme (PTAS), arbitrarily close to the optimal solution, for it, unless P = NP. Thus, every polynomial-time approximation algorithm achieves an [[approximation ratio]] strictly less than one.
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| There is a simple [[Randomized algorithm|randomized]] 0.5-[[approximation algorithm]]: for each vertex flip a coin to decide to which half of the partition to assign it.<ref>{{harvtxt|Mitzenmacher|Upfal|2005}}, Sect. 6.2.</ref><ref>{{harvtxt|Motwani|Raghavan|1995}}, Sect. 5.1.</ref> In expectation, half of the edges are cut edges. This algorithm can be [[derandomization|derandomized]] with the [[method of conditional probabilities]]; therefore there is a simple deterministic polynomial-time 0.5-approximation algorithm as well.<ref>{{harvtxt|Mitzenmacher|Upfal|2005}}, Sect. 6.3.</ref><ref>{{harvtxt|Khuller|Raghavachari|Young|2007}}.</ref> One such algorithm starts with an arbitrary partition of the vertices of the given graph <math>G = (V, E)</math> and repeatedly moves one vertex at a time from one side of the partition to the other, improving the solution at each step, until no more improvements of this type can be made. The number of iterations is at most <math>|E|</math> because the algorithm improves the cut by at least one edge at each step. When the algorithm terminates, at least half of the edges incident to every vertex belong to the cut, for otherwise moving the vertex would improve the cut. Therefore the cut includes at least <math>|E|/2</math> edges.
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| The polynomial-time approximation algorithm for max-cut with the best known approximation ratio is a method by Goemans and Williamson using [[semidefinite programming]] and [[randomized rounding]] that achieves an approximation ratio <math>\alpha \approx 0.878</math>, where <math>\alpha = \tfrac{2}{\pi} \min_{0 \le \theta \le \pi} \tfrac{\theta}{1 - \cos \theta}</math>.<ref>{{harvtxt|Gaur|Krishnamurti|2007}}.</ref><ref>{{harvtxt|Ausiello|Crescenzi|Gambosi|Kann|2003}}
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| </ref> It has been shown by Khot et al. that, if the [[unique games conjecture]] is true, then this is the best possible approximation ratio for maximum cut.<ref>{{harvtxt|Khot|Kindler|Mossel|O'Donnell|2007}}.</ref>
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| Without such unproven assumptions, it has been proven to be NP-hard to approximate the max-cut value with an approximation ratio better than <math>\tfrac{16}{17} \approx 0.941</math>.<ref>{{harvtxt|Håstad|2001}}</ref><ref>{{harvtxt|Trevisan|Sorkin|Sudan|Williamson|2000}}</ref>
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| ==Maximum bipartite subgraph==
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| A cut is a [[bipartite graph]]. The max-cut problem is essentially the same as the problem of finding a bipartite [[Glossary of graph theory#Subgraphs|subgraph]] with the most edges.
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| Let <math>G=(V,E)</math> be a graph and let <math>H=(V,X)</math> be a bipartite subgraph of ''G''. Then there is a partition (''S'', ''T'') of ''V'' such that each edge in ''X'' has one endpoint in ''S'' and another endpoint in ''T''. Put otherwise, there is a [[cut (graph theory)|cut]] in ''H'' such that the set of cut edges contains ''X''. Therefore there is a cut in ''G'' such that the set of cut edges is a superset of ''X''.
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| Conversely, let <math>G=(V,E)</math> be a graph and let ''X'' be a set of cut edges. Then <math>H=(V,X)</math> is a bipartite subgraph of ''H''.
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| In summary, if there is a bipartite subgraph with ''k'' edges, there is a cut with at least ''k'' cut edges, and if there is a cut with ''k'' cut edges, there is a bipartite subgraph with ''k'' edges. Therefore the problem of finding a '''maximum bipartite subgraph''' is essentially the same as the problem of finding a maximum cut.<ref>{{harvtxt|Newman|2008}}.</ref> The same results on NP-hardness, inapproximability and approximability apply to both the maximum cut problem and the maximum bipartite subgraph problem.
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| ==See also==
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| *[[Minimum cut]]
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| *[[Minimum k-cut]]
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| ==Notes==
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| {{reflist|2}}
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| ==References==
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| *{{citation
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| | last1=Ausiello | first1=Giorgio
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| | last2=Crescenzi | first2=Pierluigi
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| | last3=Gambosi | first3=Giorgio
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| | last4=Kann | first4=Viggo
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| | last5=Marchetti-Spaccamela | first5=Alberto
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| | last6=Protasi | first6=Marco
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| | title=Complexity and Approximation: Combinatorial Optimization Problems and Their Approximability Properties
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| | publisher=Springer
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| | year=2003
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| }}.
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| ::Maximum cut (optimisation version) is the problem ND14 in Appendix B (page 399).
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| *{{citation
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| | last1=Garey | first1=Michael R. | authorlink1=Michael R. Garey
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| | last2=Johnson | first2=David S. | authorlink2=David S. Johnson
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| | year = 1979
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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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| | isbn=0-7167-1045-5
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| }}.
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| ::Maximum cut (decision version) is the problem ND16 in Appendix A2.2.
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| ::Maximum bipartite subgraph (decision version) is the problem GT25 in Appendix A1.2.
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| *{{citation
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| | last1=Gaur | first1=Daya Ram
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| | last2=Krishnamurti | first2=Ramesh
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| | chapter=LP rounding and extensions
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| | title=Handbook of Approximation Algorithms and Metaheuristics
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| | editor-last=Gonzalez | editor-first=Teofilo F.
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| | publisher=Chapman & Hall/CRC
| |
| | year=2007
| |
| }}.
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| *{{citation
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| | last1=Goemans | first1=Michel X.
| |
| | last2=Williamson | first2=David P.
| |
| | title=Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming
| |
| | journal=[[Journal of the ACM]]
| |
| | volume=42
| |
| | issue=6
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| | year=1995
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| | pages=1115–1145
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| | doi=10.1145/227683.227684
| |
| }}.
| |
| *{{citation
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| | last=Hadlock | first=F.
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| | title=Finding a Maximum Cut of a Planar Graph in Polynomial Time
| |
| | journal=[[SIAM J. Comput.]]
| |
| | volume=4
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| | issue=3
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| | year=1975
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| | pages=221–225
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| | doi=10.1137/0204019
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| }}.
| |
| *{{citation
| |
| | last=Håstad | first=Johan | authorlink=Johan Håstad
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| | title=Some optimal inapproximability results
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| | journal=[[Journal of the ACM]]
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| | year=2001
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| | volume=48
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| | pages=798–859
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| | issue=4
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| | doi=10.1145/502090.502098
| |
| }}.
| |
| *{{citation
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| | last=Karp | first=Richard M. | authorlink=Richard Karp
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| | chapter=[[Reducibility among combinatorial problems]]
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| | editor1-first=R. E. | editor1-last=Miller
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| | editor2-first=J. W. | editor2-last=Thacher
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| | title=Complexity of Computer Computation
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| | publisher=Plenum Press
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| | pages=85–103
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| | year=1972
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| }}.
| |
| *{{citation
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| | last1=Khot | first1=Subhash |authorlink1=Subhash Khot
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| | last2=Kindler | first2=Guy
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| | last3=Mossel | first3=Elchanan
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| | last4=O'Donnell | first4=Ryan
| |
| | title=Optimal inapproximability results for MAX-CUT and other 2-variable CSPs?
| |
| | journal=[[SIAM Journal on Computing]]
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| | volume=37
| |
| | issue=1
| |
| | year=2007
| |
| | pages=319–357
| |
| | doi=10.1137/S0097539705447372
| |
| }}.
| |
| *{{citation
| |
| | last1=Khuller | first1=Samir
| |
| | last2=Raghavachari | first2=Balaji
| |
| | last3=Young | first3=Neal E.
| |
| | chapter=Greedy methods
| |
| | title=Handbook of Approximation Algorithms and Metaheuristics
| |
| | editor-last=Gonzalez | editor-first=Teofilo F.
| |
| | publisher=Chapman & Hall/CRC
| |
| | year=2007
| |
| }}.
| |
| *{{citation
| |
| | last1=Mitzenmacher | first1=Michael | authorlink1=Michael Mitzenmacher
| |
| | last2=Upfal | first2=Eli | authorlink2=Eli Upfal
| |
| | title=Probability and Computing: Randomized Algorithms and Probabilistic Analysis
| |
| | publisher=Cambridge
| |
| | year=2005
| |
| }}.
| |
| *{{citation
| |
| | last1=Motwani | first1=Rajeev | authorlink1=Rajeev Motwani
| |
| | last2=Raghavan | first2=Prabhakar
| |
| | title=Randomized Algorithms
| |
| | publisher=Cambridge
| |
| | year=1995
| |
| }}.
| |
| *{{citation
| |
| | last=Newman | first=Alantha
| |
| | chapter=Max cut
| |
| | title=Encyclopedia of Algorithms
| |
| | publisher=Springer
| |
| | editor-last=Kao | editor-first=Ming-Yang
| |
| | year=2008
| |
| | doi=10.1007/978-0-387-30162-4_219
| |
| | pages=1
| |
| | isbn=978-0-387-30770-1
| |
| }}.
| |
| *{{citation
| |
| | last1=Papadimitriou | first1=Christos H. | authorlink1=Christos Papadimitriou
| |
| | last2=Yannakakis | first2=Mihalis | authorlink2=Mihalis Yannakakis
| |
| | title=Optimization, approximation, and complexity classes
| |
| | journal=Journal of Computer and System Sciences
| |
| | year=1991
| |
| | volume=43
| |
| | issue=3
| |
| | pages=425–440
| |
| | doi=10.1016/0022-0000(91)90023-X
| |
| }}.
| |
| *{{citation
| |
| | last1=Trevisan | first1=Luca | authorlink=Luca Trevisan
| |
| | last2=Sorkin | first2=Gregory
| |
| | last3=Sudan | first3=Madhu
| |
| | last4=Williamson | first4=David
| |
| | title=Gadgets, Approximation, and Linear Programming
| |
| | journal=Proceedings of the 37th IEEE [[Symposium on Foundations of Computer Science]]
| |
| | year=2000
| |
| | pages=617–626
| |
| }}.
| |
| | |
| ==Further reading==
| |
| <!-- Applications of max-cut; could be used to expand the article: -->
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| * {{citation
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| |title=An application of combinatorial optimization to statistical physics and circuit layout design
| |
| |first1=Francisco|last1=Barahona
| |
| |first2=Martin|last2=Grötschel
| |
| |first3=Michael|last3=Jünger
| |
| |first4=Gerhard|last4=Reinelt
| |
| |journal=Operations Research
| |
| |volume=36
| |
| |issue=3
| |
| |year=1988
| |
| |pages=493–513
| |
| |doi=10.1287/opre.36.3.493
| |
| |jstor=170992
| |
| }}.
| |
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| ==External links==
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| * Pierluigi Crescenzi, Viggo Kann, Magnús Halldórsson, Marek Karpinski, Gerhard Woeginger (2000), [http://www.nada.kth.se/~viggo/wwwcompendium/node85.html "Maximum Cut"], in [http://www.nada.kth.se/~viggo/wwwcompendium/ "A compendium of NP optimization problems"].
| |
| * Andrea Casini, Nicola Rebagliati (2012), [http://code.google.com/p/maxcutpy/ "A Python library for solving Max Cut"]
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| | |
| [[Category:Graph theory objects]]
| |
| [[Category:Combinatorial optimization]]
| |
| [[Category:NP-complete problems]]
| |
| [[Category:Computational problems in graph theory]]
| |
If you present photography effectively, it helps you look much more properly at the globe around you. It is thus, on these grounds that compel various web service provider companies to integrate the same in their packages too. Your parishioners and certainly interested audience can come in to you for further information from the group and sometimes even approaching happenings and systems with the church. In the recent years, there has been a notable rise in the number of companies hiring Indian Word - Press developers. You can customize the appearance with PSD to Word - Press conversion .
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