Min Graph Coloring Problem¶
Background¶
Given a graph $G = (V,E)$, find the minimal number of colors k required to properly color it. A coloring is legal if:
- each vetrex ${v_i}$ is assigned with a color $k_i \in \{0, 1, ..., k-1\}$
- adajecnt vertex have different colors: for each $v_i, v_j$ such that $(v_i, v_j) \in E$, $k_i \neq k_j$. A graph which is k-colorable but not (k−1)-colorable is said to have chromatic number k. The maximum bound on the chromatic number is $D_G + 1$, where $D_G$ is the maximum vertex degree. The graph coloring problem is known to be in the NP-hard complexity class.
Solving the problem with classiq¶
Necessary Packages¶
In this demo, besides the classiq
package, we'll use the following packages:
%%capture
! pip install 'networkx[default]'
! pip install pyomo
! pip install matplotlib
Define the optimization problem¶
We encode the graph coloring with a matrix of variables X
with dimensions $k \times |V|$ using one-hot encoding, such that a $X_{ki} = 1$ means that vertex i is colored by color k.
We require that each vertex is colored by exactly one color and that 2 adjacent vertices have different colors.
import networkx as nx
import numpy as np
import pyomo.environ as pyo
def define_min_graph_coloring_model(graph, max_num_colors):
model = pyo.ConcreteModel()
nodes = list(graph.nodes())
colors = range(0, max_num_colors)
model.x = pyo.Var(colors, nodes, domain=pyo.Binary)
x_variables = np.array(list(model.x.values()))
adjacency_matrix = nx.convert_matrix.to_numpy_array(graph, nonedge=0)
adjacency_matrix_block_diagonal = np.kron(np.eye(degree_max), adjacency_matrix)
model.conflicting_color_constraint = pyo.Constraint(
expr=x_variables @ adjacency_matrix_block_diagonal @ x_variables == 0
)
@model.Constraint(nodes)
def each_vertex_is_colored(model, node):
return sum(model.x[color, node] for color in colors) == 1
def is_color_used(color):
is_color_not_used = np.prod([(1 - model.x[color, node]) for node in nodes])
return 1 - is_color_not_used
# minimize the number of colors in use
model.value = pyo.Objective(
expr=sum(is_color_used(color) for color in colors), sense=pyo.minimize
)
return model
Initialize the model with example graph¶
graph = nx.erdos_renyi_graph(5, 0.3, seed=79)
nx.draw_kamada_kawai(graph, with_labels=True)
degree_sequence = sorted((d for n, d in graph.degree()), reverse=True)
degree_max = max(degree_sequence)
max_num_colors = degree_max
coloring_model = define_min_graph_coloring_model(graph, max_num_colors)
show the resulting pyomo model¶
coloring_model.pprint()
4 Set Declarations each_vertex_is_colored_index : Size=1, Index=None, Ordered=Insertion Key : Dimen : Domain : Size : Members None : 1 : Any : 5 : {0, 1, 2, 3, 4} x_index : Size=1, Index=None, Ordered=True Key : Dimen : Domain : Size : Members None : 2 : x_index_0*x_index_1 : 15 : {(0, 0), (0, 1), (0, 2), (0, 3), (0, 4), (1, 0), (1, 1), (1, 2), (1, 3), (1, 4), (2, 0), (2, 1), (2, 2), (2, 3), (2, 4)} x_index_0 : Size=1, Index=None, Ordered=Insertion Key : Dimen : Domain : Size : Members None : 1 : Any : 3 : {0, 1, 2} x_index_1 : Size=1, Index=None, Ordered=Insertion Key : Dimen : Domain : Size : Members None : 1 : Any : 5 : {0, 1, 2, 3, 4} 1 Var Declarations x : Size=15, Index=x_index Key : Lower : Value : Upper : Fixed : Stale : Domain (0, 0) : 0 : None : 1 : False : True : Binary (0, 1) : 0 : None : 1 : False : True : Binary (0, 2) : 0 : None : 1 : False : True : Binary (0, 3) : 0 : None : 1 : False : True : Binary (0, 4) : 0 : None : 1 : False : True : Binary (1, 0) : 0 : None : 1 : False : True : Binary (1, 1) : 0 : None : 1 : False : True : Binary (1, 2) : 0 : None : 1 : False : True : Binary (1, 3) : 0 : None : 1 : False : True : Binary (1, 4) : 0 : None : 1 : False : True : Binary (2, 0) : 0 : None : 1 : False : True : Binary (2, 1) : 0 : None : 1 : False : True : Binary (2, 2) : 0 : None : 1 : False : True : Binary (2, 3) : 0 : None : 1 : False : True : Binary (2, 4) : 0 : None : 1 : False : True : Binary 1 Objective Declarations value : Size=1, Index=None, Active=True Key : Active : Sense : Expression None : True : minimize : 1 - (1 - x[0,0])*(1 - x[0,1])*(1 - x[0,2])*(1 - x[0,3])*(1 - x[0,4]) + 1 - (1 - x[1,0])*(1 - x[1,1])*(1 - x[1,2])*(1 - x[1,3])*(1 - x[1,4]) + 1 - (1 - x[2,0])*(1 - x[2,1])*(1 - x[2,2])*(1 - x[2,3])*(1 - x[2,4]) 2 Constraint Declarations conflicting_color_constraint : Size=1, Index=None, Active=True Key : Lower : Body : Upper : Active None : 0.0 : (x[0,1] + x[0,3] + x[0,4])*x[0,0] + (x[0,0] + x[0,3])*x[0,1] + x[0,3]*x[0,2] + (x[0,0] + x[0,1] + x[0,2])*x[0,3] + x[0,0]*x[0,4] + (x[1,1] + x[1,3] + x[1,4])*x[1,0] + (x[1,0] + x[1,3])*x[1,1] + x[1,3]*x[1,2] + (x[1,0] + x[1,1] + x[1,2])*x[1,3] + x[1,0]*x[1,4] + (x[2,1] + x[2,3] + x[2,4])*x[2,0] + (x[2,0] + x[2,3])*x[2,1] + x[2,3]*x[2,2] + (x[2,0] + x[2,1] + x[2,2])*x[2,3] + x[2,0]*x[2,4] : 0.0 : True each_vertex_is_colored : Size=5, Index=each_vertex_is_colored_index, Active=True Key : Lower : Body : Upper : Active 0 : 1.0 : x[0,0] + x[1,0] + x[2,0] : 1.0 : True 1 : 1.0 : x[0,1] + x[1,1] + x[2,1] : 1.0 : True 2 : 1.0 : x[0,2] + x[1,2] + x[2,2] : 1.0 : True 3 : 1.0 : x[0,3] + x[1,3] + x[2,3] : 1.0 : True 4 : 1.0 : x[0,4] + x[1,4] + x[2,4] : 1.0 : True 8 Declarations: x_index_0 x_index_1 x_index x conflicting_color_constraint each_vertex_is_colored_index each_vertex_is_colored value
Initialize classiq QAOA solver¶
Setting Up the Classiq Problem Instance¶
In order to solve the Pyomo model defined above, we use the Classiq combinatorial optimization engine. For the quantum part of the QAOA algorithm (QAOAConfig
) - define the number of repetitions (num_layers
):
from classiq import construct_combinatorial_optimization_model
from classiq.applications.combinatorial_optimization import OptimizerConfig, QAOAConfig
qaoa_config = QAOAConfig(num_layers=6, penalty_energy=10.0)
For the classical optimization part of the QAOA algorithm we define the maximum number of classical iterations (max_iteration
) and the $\alpha$-parameter (alpha_cvar
) for running CVaR-QAOA, an improved variation of the QAOA algorithm [3]:
optimizer_config = OptimizerConfig(alpha_cvar=0.3)
Lastly, we load the model, based on the problem and algorithm parameters, which we can use to solve the problem:
qmod = construct_combinatorial_optimization_model(
pyo_model=coloring_model,
qaoa_config=qaoa_config,
optimizer_config=optimizer_config,
)
We also set the quantum backend we want to execute on:
from classiq import set_execution_preferences
from classiq.execution import ClassiqBackendPreferences, ExecutionPreferences
backend_preferences = ExecutionPreferences(
backend_preferences=ClassiqBackendPreferences(backend_name="aer_simulator")
)
qmod = set_execution_preferences(qmod, backend_preferences)
from classiq import write_qmod
write_qmod(qmod, "min_graph_coloring")
Synthesizing the QAOA Circuit and Solving the Problem¶
We can now synthesize and view the QAOA circuit (ansatz) used to solve the optimization problem:
from classiq import show, synthesize
qprog = synthesize(qmod)
show(qprog)
Opening: https://platform.classiq.io/circuit/af263f77-25b7-4be2-addb-0aa7003f4af1?version=0.38.0.dev42%2Bfd36e2c41c
We now solve the problem by calling the execute
function on the quantum program we have generated:
from classiq import execute
res = execute(qprog).result()
We can check the convergence of the run:
from classiq.execution import VQESolverResult
vqe_result = res[0].value
vqe_result.convergence_graph
Optimization Results¶
We can also examine the statistics of the algorithm:
import pandas as pd
from classiq.applications.combinatorial_optimization import (
get_optimization_solution_from_pyo,
)
solution = get_optimization_solution_from_pyo(
coloring_model, vqe_result=vqe_result, penalty_energy=qaoa_config.penalty_energy
)
optimization_result = pd.DataFrame.from_records(solution)
optimization_result.sort_values(by="cost", ascending=True).head(5)
probability | cost | solution | count | |
---|---|---|---|---|
428 | 0.001 | 12.0 | [0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1] | 1 |
549 | 0.001 | 13.0 | [0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0] | 1 |
215 | 0.001 | 13.0 | [0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0] | 1 |
586 | 0.001 | 13.0 | [0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0] | 1 |
255 | 0.001 | 13.0 | [1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1] | 1 |
And the histogram:
optimization_result.hist("cost", weights=optimization_result["probability"])
array([[<Axes: title={'center': 'cost'}>]], dtype=object)
Let us plot the best solution:
import matplotlib.pyplot as plt
best_solution = optimization_result.solution[optimization_result.cost.idxmin()]
one_hot_solution = np.array(best_solution).reshape([max_num_colors, len(graph.nodes)])
integer_solution = np.argmax(one_hot_solution, axis=0)
nx.draw_kamada_kawai(
graph, with_labels=True, node_color=integer_solution, cmap=plt.cm.rainbow
)
Classical optimizer results¶
Lastly, we can compare to the classical solution of the problem:
from pyomo.common.errors import ApplicationError
from pyomo.opt import SolverFactory
solver = SolverFactory("couenne")
result = None
try:
result = solver.solve(coloring_model)
except ApplicationError:
print("Solver might have not exited normally. Try again")
coloring_model.display()
ERROR: Solver (asl) returned non-zero return code (-11)
ERROR: Solver log: Couenne 0.5.8 -- an Open-Source solver for Mixed Integer Nonlinear Optimization Mailing list: couenne@list.coin-or.org Instructions: http://www.coin-or.org/Couenne couenne:
Solver might have not exited normally. Try again Model unknown Variables: x : Size=15, Index=x_index Key : Lower : Value : Upper : Fixed : Stale : Domain (0, 0) : 0 : None : 1 : False : True : Binary (0, 1) : 0 : None : 1 : False : True : Binary (0, 2) : 0 : None : 1 : False : True : Binary (0, 3) : 0 : None : 1 : False : True : Binary (0, 4) : 0 : None : 1 : False : True : Binary (1, 0) : 0 : None : 1 : False : True : Binary (1, 1) : 0 : None : 1 : False : True : Binary (1, 2) : 0 : None : 1 : False : True : Binary (1, 3) : 0 : None : 1 : False : True : Binary (1, 4) : 0 : None : 1 : False : True : Binary (2, 0) : 0 : None : 1 : False : True : Binary (2, 1) : 0 : None : 1 : False : True : Binary (2, 2) : 0 : None : 1 : False : True : Binary (2, 3) : 0 : None : 1 : False : True : Binary (2, 4) : 0 : None : 1 : False : True : Binary Objectives: value : Size=1, Index=None, Active=True ERROR: evaluating object as numeric value: x[0,0] (object: <class 'pyomo.core.base.var._GeneralVarData'>) No value for uninitialized NumericValue object x[0,0]
ERROR: evaluating object as numeric value: value (object: <class 'pyomo.core.base.objective.ScalarObjective'>) No value for uninitialized NumericValue object x[0,0]
Key : Active : Value None : None : None Constraints: conflicting_color_constraint : Size=1 Key : Lower : Body : Upper None : 0.0 : None : 0.0 each_vertex_is_colored : Size=5 Key : Lower : Body : Upper 0 : 1.0 : None : 1.0 1 : 1.0 : None : 1.0 2 : 1.0 : None : 1.0 3 : 1.0 : None : 1.0 4 : 1.0 : None : 1.0
if result:
classical_solution = [
pyo.value(coloring_model.x[i, j])
for i in range(max_num_colors)
for j in range(len(graph.nodes))
]
one_hot_solution = np.array(classical_solution).reshape(
[max_num_colors, len(graph.nodes)]
)
integer_solution = np.argmax(one_hot_solution, axis=0)
nx.draw_kamada_kawai(
graph, with_labels=True, node_color=integer_solution, cmap=plt.cm.rainbow
)