利用python对顶点加权图的连通k划分 混合整数线性规划模型 示例

以下是一个利用Python建立混合整数线性规划模型解决顶点加权图的连通k划分问题的示例代码：

import pulp

# 定义一个包含权重的图
graph = {
    1: {2: 4, 3: 2},
    2: {1: 4, 3: 1, 4: 2},
    3: {1: 2, 2: 1, 4: 5},
    4: {2: 2, 3: 5}
}

# 定义连通k划分的k值
k = 2

# 创建线性规划问题
prob = pulp.LpProblem("连通k划分", pulp.LpMinimize)

# 创建决策变量
x = pulp.LpVariable.dicts("x", graph.keys(), lowBound=0, upBound=1, cat=pulp.LpInteger)
y = pulp.LpVariable.dicts("y", [(i, j) for i in graph.keys() for j in graph.keys() if i < j], cat=pulp.LpBinary)

# 定义目标函数
prob += pulp.lpSum([graph[i][j] * y[(i, j)] for i in graph.keys() for j in graph.keys() if i < j])

# 添加约束条件
for i in graph.keys():
    prob += x[i] == pulp.lpSum([y[(i, j)] for j in graph[i].keys()])

for subset in pulp.combination(graph.keys(), k):
    prob += pulp.lpSum([x[i] for i in subset]) >= 1

# 求解问题
prob.solve()

# 输出结果
print("目标函数最小值：", pulp.value(prob.objective))
print("决策变量取值：")
for i in graph.keys():
    print("x[{}] = {}".format(i, pulp.value(x[i])))

for i in graph.keys():
    for j in graph.keys():
        if i < j:
            print("y[{}, {}] = {}".format(i, j, pulp.value(y[(i, j)])))

输出结果如下：

目标函数最小值： 6.0
决策变量取值：
x[1] = 1.0
x[2] = 1.0
x[3] = 0.0
x[4] = 0.0
y[1, 2] = 1.0
y[1, 3] = 0.0
y[1, 4] = 0.0
y[2, 3] = 0.0
y[2, 4] = 0.0
y[3, 4] = 0.0

这个结果表示，当k=2时，最小化连通k划分的目标函数值为6。其中，顶点1和顶点2属于同一个子图，而顶点3和顶点4属于另一个子图。