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| In [[Optimization (mathematics)|optimization theory]], the '''max-flow min-cut theorem''' states that in a [[flow network]], the maximum amount of flow passing from the [[Glossary_of_graph_theory#Direction|''source'']] to the [[Glossary_of_graph_theory#Direction|''sink'']] is equal to the minimum capacity that, when removed in a specific way from the network, causes the situation that no flow can pass from the source to the sink.
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| The '''max-flow min-cut theorem''' is a special case of the [[dual problem|duality theorem]] for [[linear program]]s and can be used to derive [[Menger's theorem]] and the [[König's theorem (graph theory)|König-Egerváry Theorem]].
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| ==Definition==
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| Let <math>N = (V, E)</math> be a network (directed graph) with <math>s</math> and <math>t</math> being the source and the sink of <math>N</math> respectively.
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| : The '''capacity''' of an edge is a mapping <math>c \colon E \to \mathbb{R}^+</math>, denoted by ''c''<sub>uv</sub> or ''c''(''u'',''v''). It represents the maximum amount of flow that can pass through an edge.
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| : A '''flow''' is a mapping <math>f\colon E \to \mathbb{R}^+</math>, denoted by ''f''<sub>uv</sub> or ''f''(''u'',''v''), subject to the following two constraints:
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| :# <math>f_{uv} \le c_{uv}</math> for each <math>(u,v)\in E</math> (capacity constraint)
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| :# <math>\sum_{u:\,\,(u,v)\in E} f_{uv} = \sum_{u:\,\,(v,u)\in E} f_{vu}</math> for each <math>v \in V\setminus\{s,t\}</math> (conservation of flows).
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| : The '''value of flow''' is defined by <math>|f| = \sum_{\,\,v\in V} f_{sv} </math>, where <math>s</math> is the source of <math>N</math>. It represents the amount of flow passing from the source to the sink.
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| The ''maximum flow problem'' is to maximize | ''f'' |, that is, to route as much flow as possible from ''s'' to ''t''.
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| : An '''s-t cut''' ''C'' = (''S'',''T'') is a partition of ''V'' such that ''s''∈''S'' and ''t''∈''T''. The '''cut-set''' of ''C'' is the set {(''u'',''v'')∈''E'' | ''u''∈''S'', ''v''∈''T''}. Note that if the edges in the cut-set of ''C'' are removed, | ''f'' | = 0.
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| : The '''capacity''' of an ''s-t cut'' is defined by <math>c (S,T) = \sum_{(u,v) \in S \times T} c_{uv}</math>.
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| The ''minimum s-t cut problem'' is minimizing <math>c (S,T)</math>, that is, to determine S and T such that the capacity of the ''S-T cut'' is minimal.
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| ==Statement==
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| The max-flow min-cut theorem states
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| : '''The maximum value of an s-t flow is equal to the minimum capacity over all s-t cuts'''.
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| ==Linear program formulation==
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| The max-flow problem and min-cut problem can be formulated as two primal-dual linear programs.
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| {| border="0" rules="" cellspacing="0" cellpadding="0"
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| ! style="border: 1px solid darkgrey;"|
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| Max-flow (Primal)
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| ! style="border-top: 1px solid darkgrey; border-bottom: 1px solid darkgrey; border-right: 1px solid darkgrey;"|
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| Min-cut (Dual)
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| |-
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| | valign="top" align="left" style="border-right: 1px solid darkgrey; border-left: 1px solid darkgrey; padding: 1em;"|
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| maximize <math>|f| = \nabla_s</math>
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| | valign="top" align="left" style="border-right: 1px solid darkgrey; padding: 1em;"|
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| minimize <math>\sum_{(i, j) \in E} c_{ij} d_{ij} </math>
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| |-
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| | valign="top" style="border-left: 1px solid darkgrey; border-bottom: 1px solid darkgrey; border-right: 1px solid darkgrey; padding: 1em;"|
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| subject to
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| <math>
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| \begin{array}{rclr} f_{ij} & \leq & c_{ij} & (i, j) \in E \\
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| \sum_{j: (j, i) \in E} f_{ji} - \sum_{j: (i, j) \in E} f_{ij} & \leq & 0 & i \in V, i \neq s,t \\
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| \nabla_s + \sum_{j: (j, s) \in E} f_{js} - \sum_{j: (s, j) \in E} f_{sj} & \leq & 0 & \\
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| - \nabla_s + \sum_{j: (j, t) \in E} f_{jt} - \sum_{j: (t, j) \in E} f_{tj} & \leq & 0 & \\
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| f_{ij} & \geq & 0 & (i, j) \in E\\
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| \end{array} </math>
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| | valign="top" style="border-bottom: 1px solid darkgrey; border-right: 1px solid darkgrey; padding: 1em"|
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| subject to
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| <math>\begin{array}{rclr}
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| d_{ij} - p_i + p_j & \geq & 0 & (i, j) \in E \\
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| p_s - p_t & \geq & 1 & \\
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| p_i & \geq & 0 & i \in V \\
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| d_{ij} & \geq & 0 & (i, j) \in E
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| \end{array}</math>
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| |}
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| The equality in the '''max-flow min-cut theorem''' follows from the [[strong duality|strong duality theorem]] in [[linear programming]], which states that if the primal program has an optimal solution, ''x''*, then the dual program also has an optimal solution, ''y''*, such that the optimal values formed by the two solutions are equal.
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| ==Example==
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| [[File:max-flow min-cut example.svg|frame|right|A network with the value of flow equal to the capacity of an s-t cut]]
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| The figure on the right is a network having a value of flow of 7. The vertex in white and the vertices in grey form the subsets ''S'' and ''T'' of an s-t cut, whose cut-set contains the dashed edges. Since the capacity of the s-t cut is 7, which equals to the value of flow, the max-flow min-cut theorem tells us that the value of flow and the capacity of the s-t cut are both optimal in this network.
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| ==Application==
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| ===Generalized max-flow min-cut theorem===
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| In addition to edge capacity, consider there is capacity at each vertex, that is, a mapping ''c'': ''V''→''R''<sup>+</sup>, denoted by ''c''(''v''), such that the flow ''f'' has to satisfy not only the capacity constraint and the conservation of flows, but also the vertex capacity constraint
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| :<math>\sum_{i\in V} f_{iv} \le c(v)</math> for each <math>v\in V \setminus \{s,t\}.</math>
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| In other words, the amount of ''flow'' passing through a vertex cannot exceed its capacity. Define an ''s-t cut'' to be the set of vertices and edges such that for any path from ''s'' to ''t'', the path contains a member of the cut. In this case, the ''capacity of the cut'' is the sum the capacity of each edge and vertex in it.
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| In this new definition, the '''generalized max-flow min-cut theorem''' states that the maximum value of an s-t flow is equal to the minimum capacity of an s-t cut in the new sense.
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| ===Menger's theorem===
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| {{See also| Menger's Theorem}}
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| In the undirected edge-disjoint paths problem, we are given an undirected graph ''G'' = (''V'', ''E'') and two vertices ''s'' and ''t'', and we have to find the maximum number of edge-disjoint s-t paths in ''G''.
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| The '''Menger's theorem''' states that the maximum number of edge-disjoint s-t paths in an undirected graph is equal to the minimum number of edges in an s-t cut-set.
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| ===Project selection problem===
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| {{See also|Maximum flow problem}}
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| [[File:Max-flow min-cut project-selection.svg|thumb|right|A network formulation of the project selection problem with the optimal solution]]
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| In the project selection problem, there are <math>n</math> projects and <math>m</math> equipments. Each project <math>p_i</math> yields revenue <math>r(p_i)</math> and each equipment <math>q_j</math> costs <math>c(q_j)</math> to purchase. Each project requires a number of equipments and each equipment can be shared by several projects. The problem is to determine which projects and equipments should be selected and purchased respectively, so that the profit is maximized.
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| Let <math>P</math> be the set of projects ''not'' selected and <math>Q</math> be the set of equipments purchased, then the problem can be formulated as,
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| : <math>\max \{g\} = \sum_{i} r(p_i) - \sum_{p_i \in P} r(p_i) - \sum_{q_j \in Q} c(q_j).</math>
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| Since the first term does not depend on the choice of <math>P</math> and <math>Q</math>, this maximization problem can be formulated as a minimization problem instead, that is,
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| : <math>\min \{g'\} = \sum_{p_i \in P} r(p_i) + \sum_{q_j \in Q} c(q_j).</math>
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| The above minimization problem can then be formulated as a minimum-cut problem by constructing a network, where the source is connected to the projects with capacity <math>r(p_i)</math>, and the sink is connected by the equipments with capacity <math>c(q_j)</math>. An edge (<math>p_i</math>, <math>q_j</math>) with ''infinite'' capacity is added if project <math>p_i</math> requires equipment <math>q_j</math>. The s-t cut-set represents the projects and equipments in <math>P</math> and <math>Q</math> respectively. By the max-flow min-cut theorem, one can solve the problem as a [[maximum flow problem]].
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| The figure on the right gives a network formulation of the following project selection problem:
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| {| class="wikitable" style="text-align:center; width:500px;" border="1"
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| |-
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| ! width="20px" |
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| ! width="100px" |
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| Project <math>r(p_i)</math>
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| ! width="100px" |
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| Equipment <math>c(q_j)</math>
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| !
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| |-
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| ! 1
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| | 100 || 200
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| | align="left" style="padding-left: 1em;" |
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| Project 1 requires equipments 1 and 2.
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| |-
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| ! 2
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| | 200 || 100
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| | align="left" style="padding-left: 1em;" |
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| Project 2 requires equipment 2.
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| |-
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| ! 3
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| | 150 || 50
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| | align="left" style="padding-left: 1em;" |
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| Project 3 requires equipment 3.
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| |}
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| The minimum capacity of a s-t cut is 250 and the sum of the revenue of each project is 450; therefore the maximum profit ''g'' is 450 − 250 = 200, by selecting projects <math>p_2</math> and <math>p_3</math>.
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| The idea here is to 'flow' the project profits through the 'pipes' of the equipment. If we cannot fill the pipe, the equipment's return is less than its cost, and the min cut algorithm will find it cheaper to cut the project's profit edge instead of the equipment's cost edge.
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| ==History==
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| The '''max-flow min-cut theorem''' was proven by [[Peter Elias|P. Elias]], A. Feinstein, and [[C.E. Shannon]] in 1956{{ref|P. Elias, A. Feinstein, and C. E. Shannon, A note on the maximum flow through a network, IRE. Transactions on Information Theory, 2, 4 (1956), 117–119}}, and independently also by [[L.R. Ford, Jr.]] and [[D.R. Fulkerson]] in the same year{{ref|P. Elias, A. Feinstein, and C. E. Shannon, A note on the maximum flow through a network, IRE. Transactions on Information Theory, 2, 4 (1956), 117–119}}.
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| ==See also==
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| * [[Linear programming]]
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| * [[Maximum flow]]
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| * [[Minimum cut]]
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| * [[Flow network]]
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| * [[Edmonds-Karp algorithm]]
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| ==References==
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| {{reflist}}
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| # {{cite book | author = Eugene Lawler | authorlink = Eugene Lawler | title = Combinatorial Optimization: Networks and Matroids | chapter = 4.5. Combinatorial Implications of Max-Flow Min-Cut Theorem, 4.6. Linear Programming Interpretation of Max-Flow Min-Cut Theorem | year = 2001 | publisher = Dover | isbn = 0-486-41453-1 | pages = 117–120}}
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| # {{cite book | author = [[Christos H. Papadimitriou]], [[Kenneth Steiglitz]] | title = Combinatorial Optimization: Algorithms and Complexity | chapter = 6.1 The Max-Flow, Min-Cut Theorem | year = 1998| publisher = Dover | isbn = 0-486-40258-4 | pages = 120–128}}
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| # {{cite book | author = [[Vijay Vazirani|Vijay V. Vazirani]] | title = Approximation Algorithms | chapter = 12. Introduction to LP-Duality | year = 2004 | publisher = Springer | isbn = 3-540-65367-8 | pages = 93–100}}
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| [[Category:Combinatorial optimization]]
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| [[Category:Theorems in graph theory]]
| |
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