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Project # 5 Minimum Cut Problems

minimum cut definition

With respect to networks, a cut is a set of cut in two parts which are disjoint from the set of vertices, V1 and V2, located at network, leaving the source and one sink in the other.

Definition cut capacity

capacity is called a cut to the sum: S

capacity (v, w) Є v V1, V1
ЄV2 w is the part that contains the source
V2 is the part that contains the sink

Let F a flow in G and let (P, P) a cut in G. Then the ability of (p, p) is greater than or equal to the value of F, ie:

If Cij JЄP ЄP S ³ S i F i
notation If
means the sum over all vertices i
Demonstration: Watch ЄP Sj Sj S iЄP ЄP CJI = F ij S iЄP

For each side of the equation is simply the sum of Fij over all i, j Є P
Now


S i F i = Sj-Sj Fji ЄP Sj Sj ЄP

= Sj + Sj ЄP SiЄP Fji ЄP SiЄP Fji-= Sj ЄP SiЄP Fji

= Sj ЄP SiЄP Fji -Si-£ ЄP SiЄP Fji ЄP SiЄP Sj Sj £ Fji ЄP SiЄP CJI

minimal cut gives us the minimum capacity cut made in the graph.

For the calculation of minimal cut capacity does not take into account the capabilities of the cutting edges whose direction is contrary to the direction of flow.

PEAK FLOW THEOREM AND THE MINIMUM CUT

Let F be a flow in G and let (P, P) a cut in G if equality holds then the flow is maximum and the cut is minimal if and only if:

1) IF J = J for i ЄP CI, J Є P

2) Fij = 0 for i Є P, J P Є

The maximal flow value of a network equals the minimal capacity cut that can be applied to the network.

can be obtained, so the minimal cut of a network, knowing the maximal flow network obtained by applying the algorithm defined above.