Max Colorable Induced Subgraph Problem
Background
Given a graph \(G = (V,E)\) and number of colors K, find the largest induced subgraph that can be colored using up to K colors.
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 \((v_i, v_j) \in E\), \(k_i \neq k_j\).
An induced subgraph of a graph \(G = (V,E)\) is a graph \(G'=(V', E')\) such that \(V'\subset V\) and \(E' = \{(v_1, v_2) \in E\ |\ v_1, v_2 \in V'\}\).
Define the optimization problem
import networkx as nx
import numpy as np
import pyomo.environ as pyo
def define_max_k_colorable_model(graph, K):
model = pyo.ConcreteModel()
nodes = list(graph.nodes())
colors = range(0, K)
# each x_i states if node i belongs to the cliques
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(K), adjacency_matrix)
# constraint that 2 nodes sharing an edge mustn't have the same color
model.conflicting_color_constraint = pyo.Constraint(
expr=x_variables @ adjacency_matrix_block_diagonal @ x_variables == 0
)
# each node should be colored
@model.Constraint(nodes)
def each_node_is_colored_once_or_zero(model, node):
return sum(model.x[color, node] for color in colors) <= 1
def is_node_colored(node):
is_colored = np.prod([(1 - model.x[color, node]) for color in colors])
return 1 - is_colored
# maximize the number of nodes in the chosen clique
model.value = pyo.Objective(
expr=sum(is_node_colored(node) for node in nodes), sense=pyo.maximize
)
return model
Initialize the model with parameters
graph = nx.erdos_renyi_graph(6, 0.5, seed=7)
nx.draw_kamada_kawai(graph, with_labels=True)
NUM_COLORS = 2
coloring_model = define_max_k_colorable_model(graph, NUM_COLORS)
print the resulting pyomo model
coloring_model.pprint()
4 Set Declarations
each_node_is_colored_once_or_zero_index : Size=1, Index=None, Ordered=Insertion
Key : Dimen : Domain : Size : Members
None : 1 : Any : 6 : {0, 1, 2, 3, 4, 5}
x_index : Size=1, Index=None, Ordered=True
Key : Dimen : Domain : Size : Members
None : 2 : x_index_0*x_index_1 : 12 : {(0, 0), (0, 1), (0, 2), (0, 3), (0, 4), (0, 5), (1, 0), (1, 1), (1, 2), (1, 3), (1, 4), (1, 5)}
x_index_0 : Size=1, Index=None, Ordered=Insertion
Key : Dimen : Domain : Size : Members
None : 1 : Any : 2 : {0, 1}
x_index_1 : Size=1, Index=None, Ordered=Insertion
Key : Dimen : Domain : Size : Members
None : 1 : Any : 6 : {0, 1, 2, 3, 4, 5}
1 Var Declarations
x : Size=12, 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
(0, 5) : 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
(1, 5) : 0 : None : 1 : False : True : Binary
1 Objective Declarations
value : Size=1, Index=None, Active=True
Key : Active : Sense : Expression
None : True : maximize : 1 - (1 - x[0,0])*(1 - x[1,0]) + 1 - (1 - x[0,1])*(1 - x[1,1]) + 1 - (1 - x[0,2])*(1 - x[1,2]) + 1 - (1 - x[0,3])*(1 - x[1,3]) + 1 - (1 - x[0,4])*(1 - x[1,4]) + 1 - (1 - x[0,5])*(1 - x[1,5])
2 Constraint Declarations
conflicting_color_constraint : Size=1, Index=None, Active=True
Key : Lower : Body : Upper : Active
None : 0.0 : (x[0,1] + x[0,2] + x[0,4])*x[0,0] + (x[0,0] + x[0,2] + x[0,3] + x[0,5])*x[0,1] + (x[0,0] + x[0,1] + x[0,3] + x[0,4] + x[0,5])*x[0,2] + (x[0,1] + x[0,2] + x[0,4])*x[0,3] + (x[0,0] + x[0,2] + x[0,3] + x[0,5])*x[0,4] + (x[0,1] + x[0,2] + x[0,4])*x[0,5] + (x[1,1] + x[1,2] + x[1,4])*x[1,0] + (x[1,0] + x[1,2] + x[1,3] + x[1,5])*x[1,1] + (x[1,0] + x[1,1] + x[1,3] + x[1,4] + x[1,5])*x[1,2] + (x[1,1] + x[1,2] + x[1,4])*x[1,3] + (x[1,0] + x[1,2] + x[1,3] + x[1,5])*x[1,4] + (x[1,1] + x[1,2] + x[1,4])*x[1,5] : 0.0 : True
each_node_is_colored_once_or_zero : Size=6, Index=each_node_is_colored_once_or_zero_index, Active=True
Key : Lower : Body : Upper : Active
0 : -Inf : x[0,0] + x[1,0] : 1.0 : True
1 : -Inf : x[0,1] + x[1,1] : 1.0 : True
2 : -Inf : x[0,2] + x[1,2] : 1.0 : True
3 : -Inf : x[0,3] + x[1,3] : 1.0 : True
4 : -Inf : x[0,4] + x[1,4] : 1.0 : True
5 : -Inf : x[0,5] + x[1,5] : 1.0 : True
8 Declarations: x_index_0 x_index_1 x_index x conflicting_color_constraint each_node_is_colored_once_or_zero_index each_node_is_colored_once_or_zero 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=8)
qmod = combi.get_model()
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://nightly.platform.classiq.io/circuit/2zp7rUct3AEXP94PXeNWwoZkiVI
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=50, quantile=0.7)
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. The optimization is always defined as a minimzation problem, so the positive maximization objective was tranlated to a negative minimization one by the Pyomo to qmod translator.
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 | |
|---|---|---|---|
| 1153 | {'x': [[0, 1, 0, 0, 1, 0], [1, 0, 0, 1, 0, 1]]} | 0.000488 | -5 |
| 546 | {'x': [[0, 1, 0, 0, 1, 0], [1, 0, 0, 0, 0, 1]]} | 0.000488 | -4 |
| 500 | {'x': [[1, 0, 0, 1, 0, 1], [0, 0, 1, 0, 0, 0]]} | 0.000488 | -4 |
| 1170 | {'x': [[0, 1, 0, 0, 1, 0], [1, 0, 0, 1, 0, 0]]} | 0.000488 | -4 |
| 1070 | {'x': [[1, 0, 0, 0, 0, 1], [0, 1, 0, 0, 1, 0]]} | 0.000488 | -4 |
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 best solution:
import matplotlib.pyplot as plt
best_solution = optimization_result.solution[optimization_result.cost.idxmin()]["x"]
one_hot_solution = np.array(best_solution).reshape([NUM_COLORS, len(graph.nodes)])
integer_solution = np.argmax(one_hot_solution, axis=0)
colored_nodes = np.array(graph.nodes)[one_hot_solution.sum(axis=0) != 0]
colors = integer_solution[colored_nodes]
pos = nx.kamada_kawai_layout(graph)
nx.draw(graph, pos=pos, with_labels=True, alpha=0.3, node_color="k")
nx.draw(graph.subgraph(colored_nodes), pos=pos, node_color=colors, 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.