Network Flow Graph

Max flow problem Deformation:
Multi-sink and multi-source:
Add a super Source Vertex s and a super sink vertex t
S creates an edge with the capacity corresponding to the maximum outbound capacity from each source point
Each sink point to T creates an edge with the capacity corresponding to the maximum inbound capacity
Undirected graph:
Split an edge of an undirected graph into two backward directed edges with equal capacity.
Traffic limit on the vertex:
Split each vertex into two vertices, one in and one out, and then input-> output connects an edge with a capacity of vertex traffic limit C.
Minimum traffic limit:
Minimum Cost flow problem Deformation:
Similar to the maximum stream

Minimum weight matching problem:
The relationship between the two types of objects. The two objects are considered as vertices, And the link weight between them is the edge corresponding to the cost, which is converted to the minimum weight matching problem.
You can use the minimum cost stream to solve the problem.
For example, if a person wants to escape from building a to Building B, there is another ai person in building a. the maximum capacity of B is bi. AI-> BJ has a time cost and asks the minimum cost.
1. Set the vertex set of Building A to U and B to V, and add S and T points at the same time.
2. S-> UI build an edge with a capacity of AI and a cost of 0
3. Vi-> T: Build an edge with a capacity of Bi and a cost of 0
4. Vi-> VJ sets the traffic to INF, and the cost is the edge of the corresponding cost.
5. If the total number of all buildings is f, the minimum charge flow for the traffic is F.

Max closure diagram:
Closed graph: A point set of a directed graph. All outbound edges of the vertex set point to this point set.
The largest closed graph is the vertex weight and the largest of all vertex sets. It usually reflects that an event is another necessary condition and is dependent on each other.
This problem can be solved using the minimal cut.
1. Add Source Vertex S and sink vertex t
2. Change the size of all existing edges to INF.
3. For all vertices with a positive weight, establish S-> the vertex capacity is the edge of the weight value.
4. For all vertices whose weights are negative, create the vertex-> T capacity is the opposite number of edges of the weights
5. Find a minimum cut.
Minimum vertex weight overwrite set in a bipartite graph:
Point Coverage: all edges have at least one vertex in this set.
Minimum Point Weight overwrite set: Weight Value and minimum point Overwrite
1. Add Source Vertex S and sink vertex t
2. Change the size of all existing edges to INF.
3. S-> U creates an edge with a capacity of the corresponding vertex weight.
4. V-> T creates an edge with a capacity of the corresponding vertex weight
5. Find a minimum cut.

Maximum vertex independence set in a bipartite graph:
Vertex independent set: For all edges, the two endpoints of each edge are not in this set.
Maximum vertex independence set: the largest vertex.
Complementary to the minimum vertex weight overwrite set, the minimum vertex weight overwrite set is obtained, and then all vertex weights and the minimum vertex weight overwrite is the maximum independent point set.

Network Flow Graph