Maximum matching of bipartite graphs

For HungaryAlgorithmSummary:

The value of the Hungary algorithm here refers to the algorithm for finding the optimal matching of a bipartite graph. In some materials, this algorithm is called the KM algorithm, and the Hungary algorithm refers to the algorithm for finding the maximum matching.

The Hungarian algorithm looks complicated, but in summary it does the following:

Based on this theorem:

If all the subgraphs (called equal subgraphs) that meet the conditions of a [I] + B [J] = W [I, j] edge (I, j) if there is a complete match, this complete match is the maximum weight match of the Bipartite Graph. [4]

Then, we can find a complete match in an equal subgraph. If no match is found, there are two improvement schemes: 1) modify the equal subgraph and re-allocate the label; 2) expand the number of matching edges through the augmented path.

Then, continue to find the complete match. If no matching is found, continue to find it. It will always be found and it will be over.

 

References:

 

Read the following three articles:Article, Can be well understood, the maximum matching problem of the Bipartite Graph, KM algorithm, Hungary Algorithm

(1) Wikipedia's description of Hungary algorithms and references are of great significance

Http://en.wikipedia.org/wiki/Hungarian_algorithm

(2) the related concepts are clearly explained, and there is no particular Bipartite Graph

Http://people.scs.carleton.ca /~ Maheshwa/courses/573/talk01/bipartitematching. Ps

(3) Bipartite Graph Matching and Hungary algorithm.

However, there is a problem in this document. It has always said that we are looking for perfect matching. His definition of perfect matching is that each point must be matched, however, the number of vertices in the two parts of a bipartite graph must be equal. In fact, perfect matching does not need to be read from other materials. Complete matching is enough. complect matching requires that one of the two parts can match all vertices.

Http://www.cse.ust.hk /~ Golin/comp572/Notes/matching.pdf

(4) Baidu encyclopedia

Http://baike.baidu.com/view/739278.htm