Skip to main content

View on GitHub

Open this notebook in GitHub to run it yourself
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)G=(V,E), an independent set I⊆VI \subseteq V is a set of vertices such that no two vertices in II are adjacent. The Maximum Independent Set Problem is the problem of finding the independent set II with maximum cardinality. In binary form, each vertex vv is represented as being in or out of the independent set II by a binary variable xvx_v, with xv=1x_v = 1 if v∈Iv \in I, and xv=0x_v = 0 otherwise. The problem can then be formulated as Maximize ∑v∈Vxv\sum_{v \in V} x_v subject to xu+xv≤1,∀(u,v)∈Ex_{u} + x_{v} \leq 1, \forall (u, v) \in E where each xv∈0,1x_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
  • 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)
output 
mis_model.pprint()
Output:
1 Var Declarations
      x : Size=8, Index={0, 1, 2, 3, 4, 5, 6, 7}
          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={(0, 7), (2, 4), (1, 2), (0, 4), (3, 4), (1, 5), (1, 4), (0, 6), (0, 2), (2, 6), (5, 6), (3, 6), (2, 5), (3, 5)}, 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

  3 Declarations: x 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()

​
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)
Output:

Quantum program link: https://platform.classiq.io/circuit/39FhfvIzGJnBrxpW2A41Nyl81zx
Output:
https://platform.classiq.io/circuit/39FhfvIzGJnBrxpW2A41Nyl81zx?login=True&version=17
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 α\alpha-parameter (quantile) for running CVaR-QAOA, an improved variation of QAOA [3]: 
optimized_params = combi.optimize(maxiter=60, quantile=0.7)
Check the convergence of the run: 
plt.plot(combi.cost_trace)
plt.xlabel("Iterations")
plt.ylabel("Cost")
plt.title("Cost convergence")
Output:

Text(0.5, 1.0, 'Cost convergence')
output

​
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)
solutionprobabilitycost
0{‘x’: [0, 1, 0, 1, 0, 0, 0, 1]}0.025391-3
8{‘x’: [0, 0, 0, 0, 1, 1, 0, 1]}0.017578-3
35{‘x’: [1, 1, 0, 1, 0, 0, 0, 0]}0.007812-3
79{‘x’: [0, 1, 0, 0, 0, 0, 1, 1]}0.004395-3
108{‘x’: [0, 0, 1, 1, 0, 0, 0, 1]}0.002930-3
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)
output 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))
Output:

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",
)
output 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))
Output:

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",
)
output

​
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.