利用混合整数线性规划模型对加权图的连通k划分 python

连通k划分是一个经典的图分割问题，可以使用混合整数线性规划模型（MILP）进行求解。下面是一个用Python实现的例子：

首先，我们需要引入相关的库，包括Gurobi、NetworkX和Matplotlib。其中，Gurobi是一个商业数学优化软件，需要安装并获取许可证才能使用。NetworkX是一个用于创建、操作和研究复杂网络的Python库。Matplotlib是Python中的一个绘图库，用于绘制图形。

import gurobipy as gp
from gurobipy import GRB
import networkx as nx
import matplotlib.pyplot as plt

接下来，我们定义一个函数，用于生成加权图。该函数使用NetworkX库中的函数生成一个随机图，并为每条边分配一个随机权重。

def generate_weighted_graph(n_nodes, n_edges):
    G = nx.gnm_random_graph(n_nodes, n_edges)
    for (u, v) in G.edges():
        G[u][v]['weight'] = np.random.randint(0, 10)
    return G

然后，我们定义一个函数，用于解决连通k划分问题。该函数使用Gurobi库中的混合整数线性规划模型进行求解。具体来说，我们使用0-1变量$x_{ij}$表示节点$i$是否分配到集合$j$中，使用0-1变量$y_{uv}$表示边$uv$是否跨越两个不同的集合。我们的目标是最小化划分后跨越两个不同集合的边的权重之和。同时，我们需要保证每个集合都是连通的。

def solve_connected_k_partition(G, k):
    n_nodes = G.number_of_nodes()
    
    # Create model
    model = gp.Model("connected_k_partition")
    
    # Create variables
    x = {}
    y = {}
    for i in range(n_nodes):
        for j in range(k):
            x[i, j] = model.addVar(vtype=GRB.BINARY, name="x_%s_%s" % (i, j))
    for (u, v) in G.edges():
        for j in range(k):
            y[u, v, j] = model.addVar(vtype=GRB.BINARY, name="y_%s_%s_%s" % (u, v, j))
    
    # Set objective
    obj = gp.quicksum(G[u][v]['weight'] * y[u, v, j] for (u, v) in G.edges() for j in range(k))
    model.setObjective(obj, GRB.MINIMIZE)
    
    # Add constraints
    for i in range(n_nodes):
        model.addConstr(gp.quicksum(x[i, j] for j in range(k)) == 1)
    for j in range(k):
        model.addConstr(gp.quicksum(x[i, j] for i in range(n_nodes)) >= 1)
    for (u, v) in G.edges():
        for j in range(k):
            model.addConstr(y[u, v, j] <= x[u, j])
            model.addConstr(y[u, v, j] <= x[v, j])
        model.addConstr(gp.quicksum(y[u, v, j] for j in range(k)) >= 1)
    
    # Optimize model
    model.optimize()
    
    # Extract solution
    partition = []
    for j in range(k):
        subgraph = G.subgraph([i for i in range(n_nodes) if x[i, j].x > 0.5])
        partition.append(subgraph)
    
    return partition

最后，我们定义一个主函数，用于生成加权图并解决连通k划分问题。我们首先生成一个加权图，然后使用solve_connected_k_partition函数对其进行划分，并将结果绘制出来。

def main():
    n_nodes = 30
    n_edges = 100
    k = 3
    
    G = generate_weighted_graph(n_nodes, n_edges)
    
    partition = solve_connected_k_partition(G, k)
    
    pos = nx.spring_layout(G)
    colors = ['r', 'b', 'g', 'c', 'm', 'y', 'k']
    for i in range(k):
        nx.draw_networkx(partition[i], pos, node_color=colors[i], with_labels=True)
    plt.show()

完整代码如下：

import gurobipy as gp
from gurobipy import GRB
import networkx as nx
import matplotlib.pyplot as plt
import numpy as np

def generate_weighted_graph(n_nodes, n_edges):
    G = nx.gnm_random_graph(n_nodes, n_edges)
    for (u, v) in G.edges():
        G[u][v]['weight'] = np.random.randint(0, 10)
    return G

def solve_connected_k_partition(G, k):
    n_nodes = G.number_of_nodes()
    
    # Create model
    model = gp.Model("connected_k_partition")
    
    # Create variables
    x = {}
    y = {}
    for i in range(n_nodes):
        for j in range(k):
            x[i, j] = model.addVar(vtype=GRB.BINARY, name="x_%s_%s" % (i, j))
    for (u, v) in G.edges():
        for j in range(k):
            y[u, v, j] = model.addVar(vtype=GRB.BINARY, name="y_%s_%s_%s" % (u, v, j))
    
    # Set objective
    obj = gp.quicksum(G[u][v]['weight'] * y[u, v, j] for (u, v) in G.edges() for j in range(k))
    model.setObjective(obj, GRB.MINIMIZE)
    
    # Add constraints
    for i in range(n_nodes):
        model.addConstr(gp.quicksum(x[i, j] for j in range(k)) == 1)
    for j in range(k):
        model.addConstr(gp.quicksum(x[i, j] for i in range(n_nodes)) >= 1)
    for (u, v) in G.edges():
        for j in range(k):
            model.addConstr(y[u, v, j] <= x[u, j])
            model.addConstr(y[u, v, j] <= x[v, j])
        model.addConstr(gp.quicksum(y[u, v, j] for j in range(k)) >= 1)
    
    # Optimize model
    model.optimize()
    
    # Extract solution
    partition = []
    for j in range(k):
        subgraph = G.subgraph([i for i in range(n_nodes) if x[i, j].x > 0.5])
        partition.append(subgraph)
    
    return partition

def main():
    n_nodes = 30
    n_edges = 100
    k = 3
    
    G = generate_weighted_graph(n_nodes, n_edges)
    
    partition = solve_connected_k_partition(G, k)
    
    pos = nx.spring_layout(G)
    colors = ['r', 'b', 'g', 'c', 'm', 'y', 'k']
    for i in range(k):
        nx.draw_networkx(partition[i], pos, node_color=colors[i], with_labels=True)
    plt.show()

if __name__ == '__main__':
    main()