Max Independent Set
In the Maximum Independent Set Problem [1], the challenge is 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 design, 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, each vertex \(v\) is represented as 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
Go through the steps of solving the problem with the Classiq platform, using the Quantum Approximate Optimization Algorithm (QAOA) [2]. The solution is based on defining a Pyomo model for the optimization problem 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 Graph Input
Define the Pyomo model to use on the Classiq platform, with 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 included in the set. -
Constraint rule - for each edge, at least one of the corresponding node variables is 0.
-
Objective rule – the sum of the variables equals 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
To solve the Pyomo model defined above, use the CombinatorialProblem Python class. Under the hood, it translates the Pyomo model to a quantum model of QAOA, with a cost Hamiltonian translated from the Pyomo model. Choose the number of layers for the QAOA ansatz using the num_layers argument:
from classiq import *
from classiq.applications.combinatorial_optimization import CombinatorialProblem
combi = CombinatorialProblem(pyo_model=mis_model, num_layers=3)
qmod = combi.get_model()
write_qmod(qmod, "max_independent_set")
Synthesizing the QAOA Circuit and Solving the Problem
Synthesize and view the QAOA circuit (ansatz) used to solve the optimization problem:
qprog = combi.get_qprog()
show(qprog)
Opening: https://platform.classiq.io/circuit/2v4c6BmZR7F3HBXDnOj9ZWVDECt?login=True&version=0.73.0
Solve the problem by calling the optimize method of the CombinatorialProblem object. For the classical optimization part of the QAOA algorithm, define the maximum number of classical iterations (maxiter) and the \(\alpha\)-parameter (quantile) for running CVaR-QAOA, an improved variation of QAOA [3]:
optimized_params = combi.optimize(maxiter=60, quantile=0.7)
Optimization Progress: 61it [01:27, 1.43s/it]
Check the convergence of the run:
plt.plot(combi.cost_trace)
plt.xlabel("Iterations")
plt.ylabel("Cost")
plt.title("Cost convergence")
Text(0.5, 1.0, 'Cost convergence')
Optimization Results
Examine the statistics of the algorithm. The optimization is always defined as a minimization problem, so the Pyomo-to-Qmod translator changes the positive maximization objective to negative minimization.
To get samples with the optimized parameters, call the sample method:
optimization_result = combi.sample(optimized_params)
optimization_result.sort_values(by="cost").head(5)
| solution | probability | cost | |
|---|---|---|---|
| 3 | {'x': [0, 1, 0, 1, 0, 0, 0, 1]} | 0.098145 | -3 |
| 13 | {'x': [0, 0, 1, 1, 0, 0, 0, 1]} | 0.005371 | -3 |
| 12 | {'x': [1, 1, 0, 1, 0, 0, 0, 0]} | 0.005859 | -3 |
| 0 | {'x': [0, 1, 1, 1, 0, 0, 0, 1]} | 0.319336 | -2 |
| 39 | {'x': [1, 0, 0, 1, 0, 0, 0, 0]} | 0.000977 | -2 |
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)
Plot the best solution:
best_solution = optimization_result.solution[optimization_result.cost.idxmin()]["x"]
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: [1, 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",
)
Lastly, 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. (2014). A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028.
[3] Barkoutsos, Panagiotis Kl, et al. (2020). Improving variational quantum optimization using CVaR. Quantum 4 (2020): 256.