Cut (graph theory)
In graph theory, a cut is a partition of the vertices of a graph into two disjoint subsets. Any cut determines a cut-set, the set of edges that have one endpoint in each subset of the partition. These edges are said to cross the cut. In a connected graph, each cut-set determines a unique cut, and in some cases cuts are identified with their cut-sets rather than with their vertex partitions.
In a flow network, an s–t cut is a cut that requires the source and the sink to be in different subsets, and its cut-set only consists of edges going from the source's side to the sink's side. The capacity of an s–t cut is defined as the sum of the capacity of each edge in the cut-set.
Definition
A cut is a partition of of a graph into two subsets and.The cut-set of a cut is the set of edges that have one endpoint in and the other endpoint in.
If and are specified vertices of the graph, then an – cut is a cut in which belongs to the set and belongs to the set.
In an unweighted undirected graph, the size or weight of a cut is the number of edges crossing the cut. In a weighted graph, the value or weight is defined by the sum of the weights of the edges crossing the cut.
A bond is a cut-set that does not have any other cut-set as a proper subset.
Minimum cut
A cut is minimum if the size or weight of the cut is not larger than the size of any other cut. The illustration on the right shows a minimum cut: the size of this cut is 2, and there is no cut of size 1 because the graph is bridgeless.The max-flow min-cut theorem proves that the maximum network flow and the sum of the cut-edge weights of any minimum cut that separates the source and the sink are equal. There are polynomial-time methods to solve the min-cut problem, notably the Edmonds–Karp algorithm.
Maximum cut
A cut is maximum if the size of the cut is not smaller than the size of any other cut. The illustration on the right shows a maximum cut: the size of the cut is equal to 5, and there is no cut of size 6, or |E|, because the graph is not bipartite.In general, finding a maximum cut is computationally hard.
The max-cut problem is one of Karp's 21 NP-complete problems.
The max-cut problem is also APX-hard, meaning that there is no polynomial-time approximation scheme for it unless P = NP.
However, it can be approximated to within a constant approximation ratio using semidefinite programming.
Note that min-cut and max-cut are not dual problems in the linear programming sense, even though one gets from one problem to other by changing min to max in the objective function. The problem is the dual of the problem.