Max Independent Set¶
Introduction¶
In the Maximum Independent Set Problem [1], we need to find the largest subset of vertices in a given graph, such that no two vertices in the subset are adjacent. This is an NP-Hard problem in general graph structures, with applications in various fields such as network deign, bioinformatics, and scheduling.
Mathematical Formulation¶
Given a graph $G=(V,E)$, an independent set $I \subseteq V$ is a set of vertices such that no two vertices in $I$ are adjacent. The Maximum Independent Set Problem is the problem of finding the independent set $I$ with maximum cardinality. In binary form, we can represent each vertex $v$ being in or out of the independent set $I$ by a binary variable $x_v$, with $x_v = 1$ if $v \in I$, and $x_v = 0$ otherwise. The problem can then be formulated as:
Maximize $\sum_{v \in V} x_v$
Subject to:
$x_{u} + x_{v} \leq 1, \forall (u, v) \in E$
where each $x_v \in {0,1}$.
Solving with the Classiq platform¶
We go through the steps of solving the problem with the Classiq platform, using QAOA algorithm [2]. The solution is based on defining a pyomo model for the optimization problem we would like to solve.
from typing import cast
import networkx as nx
import numpy as np
import pyomo.core as pyo
from IPython.display import Markdown, display
from matplotlib import pyplot as plt
Building the Pyomo model from a graph input¶
We proceed by defining the pyomo model that will be used on the Classiq platform, using the mathematical formulation defined above:
import networkx as nx
import pyomo.core as pyo
def mis(graph: nx.Graph) -> pyo.ConcreteModel:
model = pyo.ConcreteModel()
model.x = pyo.Var(graph.nodes, domain=pyo.Binary)
@model.Constraint(graph.edges)
def independent_rule(model, node1, node2):
return model.x[node1] + model.x[node2] <= 1
model.cost = pyo.Objective(expr=sum(model.x.values()), sense=pyo.maximize)
return model
The model consists of:
- Index set declarations (model.Nodes, model.Arcs).
- Binary variable declaration for each node (model.x) indicating whether that node is chosen to be included in the set.
- Constraint rule - for each edge we require at least one of the corresponding node variables to be 0.
- Objective rule – the sum of the variables equals to the set size.
import networkx as nx
num_nodes = 8
p_edge = 0.4
graph = nx.fast_gnp_random_graph(n=num_nodes, p=p_edge, seed=12345)
nx.draw_kamada_kawai(graph, with_labels=True)
mis_model = mis(graph)
mis_model.pprint()
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=3)
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(max_iteration=60, alpha_cvar=0.7)
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=mis_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)
with open("max_independent_set.qmod", "w") as f:
f.write(qmod)
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)
We now solve the problem using the generated circuit by using the execute method:
from classiq import execute
res = execute(qprog).result()
We can check the convergence of the run:
from classiq.execution import VQESolverResult
vqe_result = res[1].value
vqe_result.convergence_graph
Optimization Results¶
We can also examine the statistics of the algorithm:
import pandas as pd
optimization_result = pd.DataFrame.from_records(res[0].value)
optimization_result.sort_values(by="cost", ascending=False).head(5)
And the histogram:
optimization_result.hist("cost", weights=optimization_result["probability"])
Let us plot the solution:
best_solution = optimization_result.solution[optimization_result.cost.idxmax()]
independent_set = [node for node in graph.nodes if best_solution[node] == 1]
print("Independent Set: ", independent_set)
print("Size of Independent Set: ", len(independent_set))
nx.draw_kamada_kawai(graph, with_labels=True)
nx.draw_kamada_kawai(
graph,
with_labels=True,
nodelist=independent_set,
node_color="r",
)
Comparison to a classical solver¶
Lastly, we can compare to the classical solution of the problem:
from pyomo.opt import SolverFactory
solver = SolverFactory("couenne")
solver.solve(mis_model)
classical_solution = [pyo.value(mis_model.x[i]) for i in graph.nodes]
independent_set_classical = [
node for node in graph.nodes if np.allclose(classical_solution[node], 1)
]
print("Classical Independent Set: ", independent_set_classical)
print("Size of Classical Independent Set: ", len(independent_set_classical))
nx.draw_kamada_kawai(graph, with_labels=True)
nx.draw_kamada_kawai(
graph,
with_labels=True,
nodelist=independent_set_classical,
node_color="r",
)
References¶
[1]: Max Independent Set (Wikipedia)
[2]: Farhi, Edward, Jeffrey Goldstone, and Sam Gutmann. "A quantum approximate optimization algorithm." arXiv preprint arXiv:1411.4028 (2014).
[3]: Barkoutsos, Panagiotis Kl, et al. "Improving variational quantum optimization using CVaR." Quantum 4 (2020): 256.