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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)

png

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/8da93adf-8ec5-44a4-838a-7d1f172c55db?version=0.41.0.dev39%2B79c8fd0855

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

png

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
301 0.001 3.0 [1, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0] 1
466 0.001 3.0 [0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 1, 0, 0] 1
734 0.001 12.0 [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1] 1
47 0.002 12.0 [0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1] 2
533 0.001 12.0 [0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0] 1

And the histogram:

optimization_result.hist("cost", weights=optimization_result["probability"])
array([[<Axes: title={'center': 'cost'}>]], dtype=object)

png

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
)

png

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
    )