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
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
Setting Up the Classiq Problem Instance
In order to solve the Pyomo model defined above, we use the CombinatorialProblem
python class. Under the hood it translates the Pyomo model to a quantum model of the QAOA algorithm [1], with cost hamiltonian translated from the Pyomo model. We can choose the number of layers for the QAOA ansatz using the argument num_layers
.
from classiq import *
from classiq.applications.combinatorial_optimization import CombinatorialProblem
combi = CombinatorialProblem(pyo_model=coloring_model, num_layers=6, penalty_factor=10)
qmod = combi.get_model()
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:
qprog = combi.get_qprog()
show(qprog)
Quantum program link: https://platform.classiq.io/circuit/2zJB0wcTNTkUTnKwKPK0f4G7edG
We now solve the problem by calling the optimize
method of the CombinatorialProblem
object. For the classical optimization part of the QAOA algorithm we define the maximum number of classical iterations (maxiter
) and the \(\alpha\)-parameter (quantile
) for running CVaR-QAOA, an improved variation of the QAOA algorithm [2]:
optimized_params = combi.optimize(maxiter=100, quantile=0.7)
Optimization Progress: 101it [12:59, 7.72s/it]
We can check the convergence of the run:
import matplotlib.pyplot as plt
plt.plot(combi.cost_trace)
plt.xlabel("Iterations")
plt.ylabel("Cost")
plt.title("Cost convergence")
Text(0.5, 1.0, 'Cost convergence')
Optimization Results
We can also examine the statistics of the algorithm. In order to get samples with the optimized parameters, we call the sample
method:
optimization_result = combi.sample(optimized_params)
optimization_result.sort_values(by="cost").head(5)
solution | probability | cost | |
---|---|---|---|
957 | {'x': [[0, 1, 1, 0, 1], [1, 0, 0, 0, 0], [0, 0... | 0.000488 | 3 |
1283 | {'x': [[0, 1, 1, 0, 0], [0, 0, 0, 1, 1], [1, 0... | 0.000488 | 3 |
1499 | {'x': [[1, 0, 1, 0, 0], [0, 0, 0, 1, 0], [0, 1... | 0.000488 | 3 |
376 | {'x': [[1, 0, 1, 0, 0], [0, 1, 0, 0, 0], [0, 0... | 0.000488 | 3 |
1435 | {'x': [[1, 0, 1, 0, 0], [0, 0, 0, 1, 1], [0, 1... | 0.000488 | 3 |
We will also want to compare the optimized results to uniformly sampled results:
uniform_result = combi.sample_uniform()
And compare the histograms:
optimization_result["cost"].plot(
kind="hist",
bins=50,
edgecolor="black",
weights=optimization_result["probability"],
alpha=0.6,
label="optimized",
)
uniform_result["cost"].plot(
kind="hist",
bins=50,
edgecolor="black",
weights=uniform_result["probability"],
alpha=0.6,
label="uniform",
)
plt.legend()
plt.ylabel("Probability", fontsize=16)
plt.xlabel("cost", fontsize=16)
plt.tick_params(axis="both", labelsize=14)
Let us plot the solution:
best_solution = optimization_result.solution[optimization_result.cost.idxmin()]
best_solution
{'x': [[1, 0, 1, 0, 0], [0, 1, 0, 0, 0], [0, 0, 0, 1, 1]]}
import matplotlib.pyplot as plt
best_solution = optimization_result.solution[optimization_result.cost.idxmin()]["x"]
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
)
References
[1]: Farhi, Edward, Jeffrey Goldstone, and Sam Gutmann. "A quantum approximate optimization algorithm." arXiv preprint arXiv:1411.4028 (2014).
[2]: Barkoutsos, Panagiotis Kl, et al. "Improving variational quantum optimization using CVaR." Quantum 4 (2020): 256.