# [Minimum edge coverage & minimum path coverage & minimum vertex coverage & maximum independent set & largest group]

Source: Internet
Author: User

Minimum edge coverage =
Maximum Independent Set = | v |-Maximum number of matches

This is done on the binary graph of the source image.
Minimum path overwrite andThe minimum edge coverage is different. N is not required for a bipartite graph.
A directed graph of x n cannot have loops, and then a bipartite graph is constructed based on the source image. The construction method is to split points into two parts. For example, I is divided into I1 and I2, And if I and j have edges, then an edge is connected between I1 and J2. The resulting Bipartite Graph

ThenMinimum path Overwrite
= N-m, n is the number of vertices in the source image, and m is the maximum matching of the new bipartite graph.
. The proof is also very simple. According to the definition of the minimum path overwrite, the same point can only belong to one path, that is, the path cannot be split, if two edges with public vertices are selected in the bipartite plot, the response is that the path has a fork in the source image. Therefore, the edges selected in the bipartite plot must have no public intersections, this is the conversion to the maximum matching.

Summary:

[Maximum number of independent undirected graphs]: selects k vertices from the V vertex, so that these k vertices are not adjacent to each other. Then the maximum K is the maximum number of independence of the graph.

[Largest group of undirected graphs]: K top is selected from the V vertices to form a complete graph. That is, either of the top two of the subgraph has direct edges.

[Minimum path overwrite (the source image is not necessarily a bipartite graph, but must be a directed graph. Split points to construct a bipartite graph)]: Find some paths in the graph to overwrite all vertices in the graph, and any vertex has only one path associated with it. Minimum path overwrite = | v |-Maximum number of matches

[Minimum edge coverage (the original image is a bipartite graph)]: Find some edges in the graph to overwrite all vertices in the graph, and any vertex has only one edge associated with it. Minimum edge coverage =
Maximum Independent Set = | v |-Maximum number of matches

[Minimum vertex overwrite]: associate each edge with at least one of the vertices with the least vertices (points in the left and right sets.

PS: these numbers have been sorted out during binary matching. There are many relationships between them. For example, the minimum overwrite count + the Maximum Independent Number = the number of vertices. Although they are all NP problems. However, there are good algorithms for special graphs, such:

In the bipartite graph, the minimum overwrite number is equal to the maximum number of matches, while the maximum number of independence is equal to the number of vertices minus the minimum overwrite number (= the maximum number of matches ), so we can use Hungary to find the maximum number of independence and so on.

A. Point Coverage set: A point set of undirected graph G, so that all edges in the graph have at least one endpoint in the set.

B. vertex independent set: A point set of undirected graph G, so that any two points in the set are not adjacent to the source image.

C. Minimum Point Coverage set: the least point coverage set in undirected graph G.

D. Maximum vertex independence set: the vertex independence set with the most points in undirected graph G.

E. Least vertex weight overwrite set: the vertex overwrite set with the smallest sum of vertex weights in an undirected graph with vertex weight.

F. Maximum vertex weight independent set: vertices with the largest sum of vertex weights in an undirected graph with vertex Weights

Nature:

Largest group = the largest independent set of the supplementary Graph

Minimum edge coverage = Maximum Independent Set of a bipartite graph = | v |-minimum path coverage

Minimum path overwrite = | v |-Maximum number of matches

Minimum vertex overwrite = maximum number of matches

Minimum vertex overwrite + maximum number of independence = | v |

Minimum Cut = minimum vertex weight overwrite set = vertex weight-maximum vertex weight Independent Set

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