混合整数线性规划模型 加权图的连通k划分 python例子

以下是一个混合整数线性规划模型的加权图的连通k划分Python例子，该例子使用PuLP库实现：

from pulp import *

# 定义图的节点和边
nodes = [1, 2, 3, 4, 5, 6]
edges = [(1, 2, 3), (1, 3, 4), (2, 3, 5), (2, 4, 6), (3, 4, 7), (4, 5, 8), (4, 6, 9), (5, 6, 10)]

# 定义连通k划分的参数
k = 2

# 定义模型
model = LpProblem("Graph Partitioning", LpMinimize)

# 定义变量
x = LpVariable.dicts("x", edges, cat=LpBinary)
y = LpVariable.dicts("y", nodes, cat=LpBinary)
z = LpVariable.dicts("z", range(k), cat=LpBinary)

# 定义目标函数
model += lpSum([x[i] * edges[i][2] for i in range(len(edges))])

# 定义约束条件
for i in range(len(nodes)):
    model += lpSum([x[j] for j in range(len(edges)) if nodes[i] in edges[j]]) <= 2 * y[nodes[i]]
    model += lpSum([x[j] for j in range(len(edges)) if nodes[i] in edges[j]]) >= y[nodes[i]]
for i in range(k):
    model += lpSum([y[j] for j in nodes]) == z[i]
for i in range(len(edges)):
    model += x[i] <= z[0]
    model += x[i] <= z[1]

# 解决模型
model.solve()

# 输出结果
print("Optimal solution:")
for i in range(len(edges)):
    if value(x[i]) == 1:
        print(edges[i])
print("Partition:")
for i in range(k):
    print([nodes[j] for j in range(len(nodes)) if value(y[nodes[j]]) == 1 and value(z[i]) == 1])

该例子中，我们定义了一个具有6个节点和8条边的加权图，并且将其划分为2个连通部分。我们使用PuLP库定义了模型、变量、目标函数和约束条件，并将其解决。最后，我们输出了解决方案和划分结果。