1 / 10

Applications of Maximum Flow and Minimum Cut Problems

Applications of Maximum Flow and Minimum Cut Problems. In this handout Transshipment problem Assignment Problem. 4. 5. D 1. B. S 1. 5. 2. 2. 5. A. 2. 15. 5. 4. 19. 9. C. S 2. D 2. Transshipment Problem. Given : Directed network G=(V, E)

macayle
Download Presentation

Applications of Maximum Flow and Minimum Cut Problems

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Applications of Maximum Flow and Minimum Cut Problems In this handout Transshipment problem Assignment Problem

  2. 4 5 D1 B S1 5 2 2 5 A 2 15 5 4 19 9 C S2 D2 TransshipmentProblem • Given: Directed network G=(V, E) Supply (source) nodes Siwith supply amounts si, i=1,…,p Demand (sink) nodes Diwith demand amounts di, i=1,…,q Total supply ≥ total demand Capacity function u: E  R • Goal: Find a feasible flow through the network which satisfies the total demand (if such a flow exists). • Ex.: 17 11 8 16

  3. 4 5 D1 B S1 5 2 2 5 T O A 2 15 5 4 19 9 C S2 D2 Solving the TransshipmentProblem via Maximum Flow • The transshipment problem can be solved by creating and solving a related instance of maximum flow problem: • Create a supersource O. For each supply node Si, add an arc O  Siwith capacity equal to the supply amount of Si. • Create a supersink T. For each demand node Di, add an arc Di  T with capacity equal to the supply amount of Di. • Find the maximum flow from O to T in the resulting auxiliary network. • If maximum flow value = total demand then the current maximum flow is a feasible flow for the transshipment problem, elsethe transshipment problem is infeasible. 17 11 17 11 16 8 8 16

  4. 4 5 D1 B S1 17 9 2 T 13 1 O 2 A 8 2 16 8 18 C S2 D2 Solving the TransshipmentProblem via Maximum Flow • In our example, • The maximum flow from O to T is obtained by applying the Augmenting Path algorithm. The maximum flow value is 25. The bold red numbers on the arcs show the flow values. • Since the maximum flow value = 25 = 17+8 = total demand, the current maximum flow is a feasible flow for the transshipment problem. 17 4 5 11 5 2 17 2 11 5 16 2 15 8 5 4 19 9 8 16

  5. B C E A D people v jobs z u x y AssignmentProblem • Given: n people and n jobs Each person can be assigned to exactly one job Each job should be assigned to exactly one person Person-job compatibility is given by a directed network (e.g., having a link A  x means “person A can do job x ”) • Goal: Find an assignment of n jobs to n people (if such an assignment exists). • Example:

  6. 1 1 1 1 1 1 1 1 1 1 B C E A D people v jobs z u x y Solving the AssignmentProblem via Maximum Flow • The assignment problem can be solved by creating and solving a related instance of the transshipment problem: • Each person-node is considered as a supply node with supply amount 1. • Each job-node is considered as a demand node with demand amount 1. • Assign capacity 1 to each arc. • Solve the resulting transshipment problem by finding maximum flow in the auxiliary network. • If maximum flow value = n then the current maximum flow gives a feasible assignment, elsethe assignment problem is infeasible.

  7. 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 B C E A D people 1 T O 1 1 1 1 v jobs z u x y 1 Solving the AssignmentProblem via Maximum Flow • In our example, the red numbers on the arcs show the optimal flow values. Since the maximum flow value is 5, the assignment problem is feasible. The feasible assignment is A  x , B  y, C  u , D  z , E  v .

  8. 1 1 1 1 1 B C E A D F 1 people 1 1 1 1 T O v w jobs 1 1 z 1 u x y 1 1 Showing the infeasibility of an assignmentproblem using maximum-flow-based arguments • Let’s solve the assignment problem below via maximum flow. The red numbers on the arcs show the optimal flow values. Since the maximum flow value = 5 < 6 = number of jobs, the assignment problem is infeasible.

  9. 1 1 1 1 1 B C E A D F 1 people 1 1 1 1 T O v w jobs 1 1 z 1 u x y 1 1 Showing the infeasibility of an assignmentproblem using minimum-cut-based arguments • The O-side of the minimum cut is {O, A, B, C, D, x, y, z} (the set of the nodes that are reachable from O via augmenting paths) • Return to the original network (delete nodes O and T and the arcs incident to them).

  10. B C E A D F people v w jobs z u x y Showing the infeasibility of an assignmentproblem using minimum-cut-based arguments • The capacity of the modified cut is 0 (since there are no arcs going from O-side to T-side). • There are 2 people (E, F) and 3 jobs (u, v, w) on T-side. • Thus, one of the 3 jobs should be assigned to a person from O-side (A, B, C, D). • But there are no arcs going from O-side to T-side. ( A, B, C, D can’t do the jobs u, v, w). • Thus, there is no way to assign the T-side jobs, and the assignment problem is infeasible.

More Related