Max-Flow-Min-Cut-Theorem

In the field of graph theory, the Max -Flow Min -Cut theorem refers to a set that gives a statement about the relationship of maximum flows and minimum cuts of a river network. The theorem states:

The theorem is a generalization of Menger's theorem. It became independent from L.R. in 1956 Ford Jr. and D. R. Fulkerson and P. Elias, A. Feinstein, and CE Shannon proved.

• 3.1 algorithm for finding minimum cuts

Definitions

Be a finite directed graph with nodes and edges. Each edge from node to node have a non-negative capacity addition, there is a source node, where the network flow begins, and a destination node in the network flow ends.

Set

The following three statements are equivalent:

Sketch of proof

• If there were an augmenting path, it might be the river along its zoom; thus the flow can not have been maximal.

• If there is no augmenting path, then divide the graph in which nodes can be reached from the residual network, and the rest Then ( it would not be 0, would have been reached ). Then, for this cut.

• If it were not maximal, so you could enlarge it. Since less than or equal to the capacity of each section, can be used for at least a section not yet taken advantage of the capacity; In addition, for any section, because otherwise there would be no path for augmenting the flow expansion and the flow would be at a maximum.

In particular, this shows that the maximum flow is equal to the minimum average: Due to the size of at least 3, it has a section, that is at least the smallest, and also due to the second maximum value that, because of the residual network already when the size of the smallest section reaches has, may contain no more augmenting path.

Example

Be the flow network given to the nodes, and a maximum flow from the source to the sink of size 5

There are three minimal cuts on this network:

Note: is not a minimal cut, though, and fully be used; because there is still an edge in the residual network (r, q) of the remaining capacity.

Algorithm for finding minimum cuts

There are various algorithms for finding minimum cuts. The following algorithm finds the edges of a minimum cut directly from the residual network and making itself the properties of the max -flow min-cut theorem advantage. The residual flow graph can be generated for example by means of the algorithm of Ford and Fulkerson.

A finite directed graph with a source, a sink, and each edge has a nonnegative capacity. findeKantenEinesMinCut () 1 Residualnetztwerk () 2 3 4 For each node 5 If path () exists in 6 then 7 otherwise 8 9 For each edge 10 If start node () and end node ( ) is 11 then 12 is now the set of edges of a minimum cut   would the cut edges of included in the example above.