**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