"Network Flow": survey design, route scheduling, project selection

1) Survey design

Problem Description:

A bunch of products and a bunch of customers, each product can only be [C, C '] customers do feedback, each customer can only do [P, p '] products feedback; Ask, can you design a survey plan, so that the above conditions can be satisfied. (The main limitation here is the requirement of the minimum flow C, p on the edge)

Method:

The flow on the edge of (s, i) indicates the number of products that the customer I can investigate and therefore has a limit of [C, C '];

The flow on the edge of (j, T) corresponds to the number of customers who are asked for product J, and therefore has a limit of [P, p '];

Each customer I to the product he bought between J has a capacity of 1 side (I, j), but no nether requirements (the Nether is 0), because of the product J, customer I do not have to make feedback (as long as between [C, C ']);

The flow on the edge of (t, s) represents the total number of questions asked, so it is [Sigma (c), Sigma (c ')];

The requirement for all nodes is 0.

finally determine whether the new map is feasible to circulate.

Correctness:

P276, theorem 7.53, the preceding structure of Figure G ' has a viable flow, when and only if there is a workable way to design this survey.

2) Route scheduling

Problem Description:

A bunch of routes four yuan (Startid, StartTime, Endid, EndTime); An aircraft can fly except for a normal flight I, Flight J that meets one of the following conditions:

1) The end point of I and the beginning of J, and I and J have enough time to maintain the aircraft;

2) I and J have enough time to fly from the end of I to the starting point of J, while maintaining the time of the aircraft (equivalent to additional flights);

Q: Given the K-planes, can we get this stack of flights done?

Method:

Note increase S to the side of t ...

finally determine whether the new map is feasible to circulate.

Correctness:

P279, theorem 7.54, there is a way to perform all flights with a K-plane, when and only the current surface structure of Figure G ' has a viable flow.

3) Project selection

Problem Description:

A pile of projects, each project P has a profit pi (pi can be negative, indicating that some of the basic projects are cost-effective, some projects are profiteering), also stipulates that some project set A is the basis of other project set B (a must be completed before you can start B); We call project set a feasible, When and only if the underlying project (set) of any of the items in a is still in the A collection.

Q: Can you find a project collection A with the largest profit, and make sure that a is feasible.

Method:

Finally, the minimum cut (a ', B ') in the new diagram is computed, and a '-{s} is declared to be the optimal set of items.

Correctness:

P286, theorem 7.58, if (a ', B ') is a minimum cut of G ', then the set a '-{s} is an optimal solution for a project selection problem.