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.
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:
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.
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()
2 Set Declarations
independent_rule_index : Size=1, Index=None, Ordered=False
Key : Dimen : Domain : Size : Members
None : 2 : Any : 14 : {(0, 2), (0, 4), (0, 6), (0, 7), (1, 2), (1, 4), (1, 5), (2, 4), (2, 5), (2, 6), (3, 4), (3, 5), (3, 6), (5, 6)}
x_index : Size=1, Index=None, Ordered=False
Key : Dimen : Domain : Size : Members
None : 1 : Any : 8 : {0, 1, 2, 3, 4, 5, 6, 7}
1 Var Declarations
x : Size=8, Index=x_index
Key : Lower : Value : Upper : Fixed : Stale : Domain
0 : 0 : None : 1 : False : True : Binary
1 : 0 : None : 1 : False : True : Binary
2 : 0 : None : 1 : False : True : Binary
3 : 0 : None : 1 : False : True : Binary
4 : 0 : None : 1 : False : True : Binary
5 : 0 : None : 1 : False : True : Binary
6 : 0 : None : 1 : False : True : Binary
7 : 0 : None : 1 : False : True : Binary
1 Objective Declarations
cost : Size=1, Index=None, Active=True
Key : Active : Sense : Expression
None : True : maximize : x[0] + x[1] + x[2] + x[3] + x[4] + x[5] + x[6] + x[7]
1 Constraint Declarations
independent_rule : Size=14, Index=independent_rule_index, Active=True
Key : Lower : Body : Upper : Active
(0, 2) : -Inf : x[0] + x[2] : 1.0 : True
(0, 4) : -Inf : x[0] + x[4] : 1.0 : True
(0, 6) : -Inf : x[0] + x[6] : 1.0 : True
(0, 7) : -Inf : x[0] + x[7] : 1.0 : True
(1, 2) : -Inf : x[1] + x[2] : 1.0 : True
(1, 4) : -Inf : x[1] + x[4] : 1.0 : True
(1, 5) : -Inf : x[1] + x[5] : 1.0 : True
(2, 4) : -Inf : x[2] + x[4] : 1.0 : True
(2, 5) : -Inf : x[2] + x[5] : 1.0 : True
(2, 6) : -Inf : x[2] + x[6] : 1.0 : True
(3, 4) : -Inf : x[3] + x[4] : 1.0 : True
(3, 5) : -Inf : x[3] + x[5] : 1.0 : True
(3, 6) : -Inf : x[3] + x[6] : 1.0 : True
(5, 6) : -Inf : x[5] + x[6] : 1.0 : True
5 Declarations: x_index x independent_rule_index independent_rule cost
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 *
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.execution import ClassiqBackendPreferences
qmod = set_execution_preferences(
qmod, backend_preferences=ClassiqBackendPreferences(backend_name="simulator")
)
write_qmod(qmod, "max_independent_set")
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 = synthesize(qmod)
show(qprog)
Opening: https://platform.classiq.io/circuit/7e58acc3-dfdb-4ebf-88c6-c980aa6e5089?version=0.41.0.dev39%2B79c8fd0855
We now solve the problem by calling the execute
function on the quantum program we have generated:
result = execute(qprog).result_value()
We can check the convergence of the run:
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(
mis_model, vqe_result=result, penalty_energy=qaoa_config.penalty_energy
)
optimization_result = pd.DataFrame.from_records(solution)
optimization_result.sort_values(by="cost", ascending=False).head(5)
probability | cost | solution | count | |
---|---|---|---|---|
35 | 0.009 | 3.0 | [0, 0, 0, 0, 1, 0, 1, 1] | 9 |
14 | 0.016 | 3.0 | [0, 0, 0, 0, 1, 1, 0, 1] | 16 |
6 | 0.023 | 3.0 | [0, 1, 0, 0, 0, 0, 1, 1] | 23 |
1 | 0.035 | 3.0 | [0, 0, 1, 1, 0, 0, 0, 1] | 35 |
92 | 0.003 | 3.0 | [1, 1, 0, 1, 0, 0, 0, 0] | 3 |
And the histogram:
optimization_result.hist("cost", weights=optimization_result["probability"])
array([[<Axes: title={'center': 'cost'}>]], dtype=object)
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))
Independent Set: [2, 3, 7]
Size of Independent Set: 3
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))
Classical Independent Set: [0, 1, 3]
Size of Classical Independent Set: 3
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.