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Minimum Cost Maximum Flow Algorithm with restrictions

**Minimum Cost Maximum Flow Algorithm with Restrictions**

This algorithm aims to connect clients to servers while minimizing the total distance traveled by the connections, subject to specific constraints.

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

Mixed-Integer Linear Programming Model

Alternatively, you can formulate this problem as a mixed-integer linear program (MIP), which can be solved using specialized software.

Variables

  • xij: Binary variable indicating if client i is connected to server j (1) or not (0)
  • yj: Number of clients connected to server j

Constants

  • dij: Distance between client i and server j
  • f: Minimum number of clients per server

Objective

Minimize total connection distance: Σi,jdijxij

Constraints

  • Each client connects to exactly one server: Σjxij = 1 for each client i
  • Number connected clients equals number assigned to server: Σixij - yj = 0 for each server j
  • Minimum client count per server: yj - f >= 0 for each server j
  • Maximum client count per server: yj - f <= 1 for each server j
  • Binary connection variables: xij = 0 or 1
This MIP model provides an optimal solution but requires specialized software to solve.

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