Minimum cut

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A graph and two of its cuts. The dotted line in red represents a cut with three crossing edges. The dashed line in green represents one of the minimum cuts of this graph, crossing only two edges.[1]

In graph theory, a minimum cut of a graph is a cut (a partition of the vertices of a graph into two disjoint subsets that are joined by at least one edge) that is minimal in some sense.

Variations of the minimum cut problem consider weighted graphs, directed graphs, terminals, and partitioning the vertices into more than two sets.

Without Terminals[edit]

The minimum cut problem in undirected, weighted graphs can be solved in polynomial time by the Stoer-Wagner algorithm. In the special case when the graph is unweighted, Karger's algorithm provides an efficient randomized method for finding the cut. In this case, the minimum cut equals the edge connectivity of the graph.

A generalization of the minimum cut problem without terminals is the minimum k-cut, in which the goal is to partition the graph into at least k connected components by removing as few edges as possible. For a fixed value of k, this problem can be solved in polynomial time, though the algorithm is not practical for large k. [2]

With Terminals[edit]

When two terminals are given, they are typically referred to as a source and sink. In a directed, weighted flow network, the minimum cut separates the source and sink vertices and minimizes the total weight on the edges that are directed from the source side of the cut to the sink side of the cut. As shown in the max-flow min-cut theorem, the weight of this cut equals the maximum amount of flow that can be sent from the source to the sink in the given network.

In a weighted, undirected network, it is possible to calculate the cut that separates a particular pair of vertices from each other and has minimum possible weight. A system of cuts that solves this problem for every possible vertex pair can be collected into a structure known as the Gomory–Hu tree of the graph.

A generalization of the minimum cut problem with terminals is the k-terminal cut, or multiterminal cut. This problem is NP-hard, even for .[3]

Applications[edit]

Graph partition problems are a family of combinatorial optimization problems in which a graph is to be partitioned into two or more parts with additional constraints such as balancing the sizes of the two sides of the cut.

Number of minimum cuts[edit]

A graph with vertices can at the most have distinct minimum cuts. This bound is tight in the sense that a (simple) cycle on vertices has exactly minimum cuts.

See also[edit]

  • Maximum cut, a cut that is as large as possible instead of being as small as possible.
  • Vertex separator, an analogous concept to minimum cuts for vertices instead of edges.

References[edit]

  1. ^ "4 Min-Cut Algorithms".
  2. ^ "A Polynomial Algorithm for the k-cut Problem for Fixed k".
  3. ^ "The Complexity of Multiterminal Cuts" (PDF).