Minimum Cost Maximum Flow Algorithm with restrictions

Minimum Cost Maximum Flow Algorithm with Restrictions

The minimum cost maximum flow algorithm with restrictions is a variant of the maximum flow problem that takes into account the cost of sending flow through edges. The goal is to find a flow that maximizes the total flow while minimizing the total cost.

Algorithm

LOOP until stop
Loop over servers
  SET closest = NULL
  SET shortest = INFINITY
  Loop over clients 
     IF client has server
         CONTINUE
     SET dist = distance from client to server
     IF dist < shortest
       SET closest = client
       SET shortest = dist
  ENDLOOP over clients
  IF closest == NULL
       STOP
  Connect closest client to server
ENDLOOP over servers
ENDLOOP until stop

Example

Consider the following example:

• There are 3 servers and 5 clients.
• The cost of sending flow from a server to a client is given by the following matrix:

| | Client 1 | Client 2 | Client 3 | Client 4 | Client 5 | |---|---|---|---|---|---| | Server 1 | 1 | 2 | 3 | 4 | 5 | | Server 2 | 6 | 7 | 8 | 9 | 10 | | Server 3 | 11 | 12 | 13 | 14 | 15 | The minimum cost maximum flow is 15 and the corresponding flow is: | Server | Client 1 | Client 2 | Client 3 | Client 4 | Client 5 | |---|---|---|---|---|---| | Server 1 | 1 | 2 | 3 | 4 | 5 | | Server 2 | 6 | 7 | 8 | 9 | 10 | | Server 3 | 11 | 12 | 13 | 14 | 15 |