 Maxflow mincut theorem

In optimization theory, the maxflow mincut theorem states that in a flow network, the maximum amount of flow passing from the source to the sink is equal to the minimum capacity which when removed in a specific way from the network causes the situation that no flow can pass from the source to the sink.
The maxflow mincut theorem is a special case of the duality theorem and can be used to derive Menger's theorem and the KönigEgerváry Theorem.
Contents
Definition
Let N = (V,E) be a network (directed graph) with s and t being the source and the sink of N respectively.
 The capacity of an edge is a mapping c: E→R^{+}, denoted by c_{uv} or c(u,v). It represents the maximum amount of flow that can pass through an edge.
 A flow is a mapping f: E→R^{+}, denoted by f_{uv} or f(u,v), subject to the following two constraints:
 for each (capacity constraint)
 for each (conservation of flows).
 The value of flow is defined by , where s is the source of N. It represents the amount of flow passing from the source to the sink.
The maximum flow problem is to maximize  f , that is, to route as much flow as possible from s to t.
 An st cut C = (S,T) is a partition of V such that s∈S and t∈T. The cutset of C is the set {(u,v)∈E  u∈S, v∈T}. Note that if the edges in the cutset of C are removed,  f  = 0.
 The capacity of an st cut is defined by .
The minimum cut problem is to minimize c(S,T), that is, to determine S and T such that the capacity of the ST cut is minimal.
Statement
The maxflow mincut theorem states
 The maximum value of an st flow is equal to the minimum capacity of an st cut.
Linear program formulation
The maxflow problem and mincut problem can be formulated as two primaldual linear programs.
Maxflow (Primal)
Mincut (Dual)
maximize
minimize
subject to
subject to
The equality in the maxflow mincut theorem follows from the strong duality theorem, 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.
Example
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 st cut, whose cutset contains the dashed edges. Since the capacity of the st cut is 7, which is equal to the value of flow, the maxflow mincut theorem tells us that the value of flow and the capacity of the st cut are both optimal in this network.
Application
Generalized maxflow mincut theorem
In addition to edge capacity, consider there is capacity at each vertex, that is, a mapping c: V→R^{+}, 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
 for each
In other words, the amount of flow passing through a vertex cannot exceed its capacity. Define an st 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.
In this new definition, the generalized maxflow mincut theorem states that the maximum value of an st flow is equal to the minimum capacity of an st cut in the new sense.
Menger's theorem
See also: Menger's TheoremIn the undirected edgedisjoint 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 edgedisjoint st paths in G.
The Menger's theorem states that the maximum number of edgedisjoint st paths in an undirected graph is equal to the minimum number of edges in an st cutset.
Project selection problem
See also: Maximum flow problemIn the project selection problem, there are n projects and m equipments. Each project p_{i} yields revenue r(p_{i}) and each equipment q_{j} costs c(q_{j}) 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.
Let P be the set of projects not selected and Q be the set of equipments purchased, then the problem can be formulated as,
Since r(p_{i}) and c(q_{j}) are positive, this maximization problem can be formulated as a minimization problem instead, that is,
The above minimization problem can then be formulated as a minimumcut problem by constructing a network, where the source is connected to the projects with capacity r(p_{i}), and the sink is connected by the equipments with capacity c(q_{j}). An edge (p_{i}, q_{j}) with infinite capacity is added if project p_{i} requires equipment q_{j}. The st cutset represents the projects and equipments in P and Q respectively. By the maxflow mincut theorem, one can solve the problem as a maximum flow problem.
The figure on the right gives a network formulation of the following project selection problem:
Project r(p_{i})
Equipment c(q_{j})
1 100 200 Project 1 requires equipments 1 and 2.
2 200 100 Project 2 requires equipment 2.
3 150 50 Project 3 requires equipment 3.
The minimum capacity of a st 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 p_{2} and p_{3}.
History
The maxflow mincut theorem was proved by P. Elias, A. Feinstein, and C.E. Shannon in 1956, and independently also by L.R. Ford, Jr. and D.R. Fulkerson in the same year.
See also
 Linear programming
 Maximum flow
 Minimum cut
 Flow network
 EdmondsKarp algorithm
References
 Eugene Lawler (2001). "4.5. Combinatorial Implications of MaxFlow MinCut Theorem, 4.6. Linear Programming Interpretation of MaxFlow MinCut Theorem". Combinatorial Optimization: Networks and Matroids. Dover. pp. 117–120. ISBN 0486414531.
 Christos H. Papadimitriou, Kenneth Steiglitz (1998). "6.1 The MaxFlow, MinCut Theorem". Combinatorial Optimization: Algorithms and Complexity. Dover. pp. 120–128. ISBN 0486402584.
 Vijay V. Vazirani (2004). "12. Introduction to LPDuality". Approximation Algorithms. Springer. pp. 93–100. ISBN 3540653678.
Categories: Combinatorial optimization
 Theorems in discrete mathematics
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