图着色 Gurobi 约束

Question

我正在尝试使用 networkx 和 gurobi 解决图形着色问题的一些限制。这是我写的所有代码：

import networkx as nx
import gurobi as gb
from itertools import combinations, chain
import pygraphviz as pygv
import os
import matplotlib.pyplot as plt
from IPython.display import SVG, display

创建图表，添加节点和边以及两个列表。

G = nx.Graph()
G.add_nodes_from ([1,2,3,4,5])
G.add_edge(1,2)
G.add_edge(1,3)
G.add_edge(1,4)
G.add_edge(1,5)
G.add_edge(2,3)
G.add_edge(2,4)
G.add_edge(3,5)
G.add_edge(4,5)
U = list(G.nodes)
K = G.number_of_edges()
Z = []

用颜色创建一个列表。我们假设 K = {0, 1, . . . , K − 1} 和 K ≤ |E|

def nrofCol():
    Z.clear()
    z = 0
    while z < K - 1:
        Z.append(z)
        z = z+1
    return Z

Z = nrofCol()

为每条边添加颜色属性

for colored_arc in ((u,v,z) for u,v in G.edges() for z in Z):
    G[colored_arc[0]][colored_arc[1]][colored_arc[2]] = colored_arc

并使用 Gurobi 向模型添加变量：

mdic = gb.Model()
indices = []

for u,v in G.edges(): 
    for z in Z:
        indices.append((u,v,z))

# binary variable that assing 1.0 to the color associated to the edge and 0.0 to the others

x = mdic.addVars(indices, vtype = gb.GRB.BINARY)

# decision variable S i for i ∈ V represents the maximum color in the set of colors assigned to edges incident to vertex i

S_i = []
for u in U:
    S_i.append(mdic.addVar(vtype=gb.GRB.CONTINUOUS, lb = G.degree[u] - 1, ub = K - 1, \
                        name = 'max_color'+str(u)))

# decision variable s_i for i ∈ V represents the minimum color in the set of colors assigned to edges incident to vertex i
s_i = []
for u in U:
    s_i.append(mdic.addVar(vtype=gb.GRB.CONTINUOUS, lb = 0.0, ub = K - G.degree[u], \
                        name='min_color'+str(u)))

mdic.update()

然后是约束：

# 1a- Guarantee that adjacent edges take different colors

for u in U:
    for z in Z: 
        mdic.addConstr(x.sum(u,'*',z) <= 1, name='different_color')


mdic.update()


# 1a- Guarantee that adjacent edges take different colors

for u in U:
    for z in Z:
        mdic.addConstr(x.sum('*',u,z) <= 1, name='different_color')


mdic.update()

# 1b- Guarantee that every edge takes exactly one color
for u,v in G.edges():
    mdic.addConstr(x.sum(u,v) == 1, name='one_color')

mdic.update()

# 1c- Enforce Si to be greater than or equal to the max color assigned to the edges incident to vertex i

expr = 0
for u,v in G.edges():
    for z in Z:       
        expr += z * x[u,v,z]
    mdic.addConstr(S_i[u] >= expr, name='max')
    expr = 0

# 1d- Enforce si to be less than or equal to the min color assigned to the edges incident to vertex i

expr = 0
for u,v in G.edges():
    for z in Z:       
        expr += z * x[u,v,z]
mdic.addConstr(s_i[u] <= expr, name='min')
expr = 0

mdic.update()

其中 Z 是可用颜色的集合。

# objective function
expr20=0
for u in U:
    expr20+=(S_i[u] - s_i[u] - G.degree[u] + 1)
mdic.setObjective(expr20, gb.GRB.MINIMIZE)

mdic.update()

mdic.optimize()

Constraints

第一个是目标函数，1a到1d是约束，其他是ub和lb。

Answer 1

假设你使用无向图，我发现了几个问题：

屏幕截图中的约束 (1a) 确保相邻边缘具有不同的颜色。但是，通过您实施此约束，可能会发生传入和传出边缘具有相同颜色的情况。例如，边 {1,3} 和 {3,5} 可以具有相同的颜色。那是因为您分别处理传入和传出边缘。作为一种解决方案，您可以将循环组合成一个：

for u in U:
    for z in Z: 
        mdic.addConstr(x.sum(u,'*',z) + x.sum('*',u,z) <= 1, name='different_color')

约束 (1c) 在您的实现中也只考虑出边。例如，S_i[5] 不会被分配，因为它只有传入边。这应该有效：

expr = 0
for u,v in G.edges():
    for z in Z:       
        expr += z * x[u,v,z]
    mdic.addConstr(S_i[u] >= expr, name='max')
    mdic.addConstr(S_i[v] >= expr, name='max')
    expr = 0

约束 (1d) 也是如此。 Morover addConstr 行在循环之外，但这可能只是一个格式错误：

expr = 0
for u,v in G.edges():
    for z in Z:       
        expr += z * x[u,v,z]
    mdic.addConstr(s_i[u] <= expr, name='min')
    mdic.addConstr(s_i[v] <= expr, name='min')
    expr = 0