Max K-Vertex Cover
Introduction
The Max K-Vertex Cover problem [1] is a classical problem in graph theory and computer science, where we aim to find a set of vertices such that each edge of the graph is incident to at least one vertex in the set, and the size of the set does not exceed a given number \(k\).
Mathematical Formulation
The Max-k Vertex Cover problem can be formulated as an Integer Linear Program (ILP):
Minimize: \(\sum_{(i,j) \in E} (1 - x_i)(1 - x_j)\)
Subject to: \(\sum_{i \in V} x_i = k\)
and \(x_i \in \{0, 1\} \quad \forall i \in V\)
Where:
-
\(x_i\) is a binary variable that equals 1 if node \(i\) is in the cover and 0 otherwise
-
\(E\) is the set of edges in the graph
-
\(V\) is the set of vertices in the graph
-
\(k\) is the maximum number of vertices allowed in the cover
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 mvc(graph: nx.Graph, k: int) -> pyo.ConcreteModel:
model = pyo.ConcreteModel()
model.x = pyo.Var(graph.nodes, domain=pyo.Binary)
model.amount_constraint = pyo.Constraint(expr=sum(model.x.values()) == k)
def obj_expression(model):
# number of edges not covered
return sum((1 - model.x[i]) * (1 - model.x[j]) for i, j in graph.edges)
model.cost = pyo.Objective(rule=obj_expression, sense=pyo.minimize)
return model
The model contains:
-
Index set declarations (model.Nodes, model.Arcs).
-
Binary variable declaration for each node (model.x) indicating whether the variable is chosen for the set.
-
Constraint rule – ensures that the set is of size k.
-
Objective rule – counts the number of edges not covered; i.e., both related variables are zero.
import networkx as nx
K = 5
num_nodes = 10
p_edge = 0.5
graph = nx.erdos_renyi_graph(n=num_nodes, p=p_edge, seed=13)
nx.draw_kamada_kawai(graph, with_labels=True)
mvc_model = mvc(graph, K)
mvc_model.pprint()
1 Set Declarations
x_index : Size=1, Index=None, Ordered=False
Key : Dimen : Domain : Size : Members
None : 1 : Any : 10 : {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}
1 Var Declarations
x : Size=10, 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
8 : 0 : None : 1 : False : True : Binary
9 : 0 : None : 1 : False : True : Binary
1 Objective Declarations
cost : Size=1, Index=None, Active=True
Key : Active : Sense : Expression
None : True : minimize : (1 - x[0])*(1 - x[1]) + (1 - x[0])*(1 - x[5]) + (1 - x[0])*(1 - x[6]) + (1 - x[0])*(1 - x[7]) + (1 - x[0])*(1 - x[8]) + (1 - x[1])*(1 - x[2]) + (1 - x[1])*(1 - x[4]) + (1 - x[1])*(1 - x[5]) + (1 - x[1])*(1 - x[6]) + (1 - x[1])*(1 - x[9]) + (1 - x[2])*(1 - x[3]) + (1 - x[2])*(1 - x[4]) + (1 - x[3])*(1 - x[6]) + (1 - x[3])*(1 - x[8]) + (1 - x[4])*(1 - x[6]) + (1 - x[4])*(1 - x[7]) + (1 - x[4])*(1 - x[8]) + (1 - x[4])*(1 - x[9]) + (1 - x[5])*(1 - x[6]) + (1 - x[7])*(1 - x[8]) + (1 - x[8])*(1 - x[9])
1 Constraint Declarations
amount_constraint : Size=1, Index=None, Active=True
Key : Lower : Body : Upper : Active
None : 5.0 : x[0] + x[1] + x[2] + x[3] + x[4] + x[5] + x[6] + x[7] + x[8] + x[9] : 5.0 : True
4 Declarations: x_index x amount_constraint 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 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=mvc_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="simulator")
)
qmod = set_execution_preferences(qmod, backend_preferences)
from classiq import write_qmod
write_qmod(qmod, "max_k_vertex_cover")
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)
Opening: https://platform.classiq.io/circuit/308cc52e-d608-47d6-9ccb-ceb9f2e932a2?version=0.41.0.dev39%2B79c8fd0855
We now solve the problem by calling the execute function on the quantum program we have generated:
from classiq import execute
res = execute(qprog).result()
We can check the convergence of the run:
from classiq.execution import VQESolverResult
vqe_result = res[0].value
vqe_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(
mvc_model, vqe_result=vqe_result, penalty_energy=qaoa_config.penalty_energy
)
optimization_result = pd.DataFrame.from_records(solution)
optimization_result.sort_values(by="cost", ascending=True).head(5)
| probability | cost | solution | count | |
|---|---|---|---|---|
| 0 | 0.058 | 1.0 | [1, 1, 0, 0, 1, 0, 1, 0, 1, 0] | 58 |
| 2 | 0.024 | 1.0 | [1, 1, 0, 1, 1, 0, 0, 0, 1, 0] | 24 |
| 272 | 0.001 | 2.0 | [0, 1, 1, 0, 1, 0, 1, 0, 1, 0] | 1 |
| 220 | 0.001 | 2.0 | [1, 1, 0, 1, 1, 1, 0, 0, 0, 0] | 1 |
| 156 | 0.002 | 2.0 | [1, 1, 0, 0, 1, 1, 0, 0, 1, 0] | 2 |
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.idxmin()]
best_solution
[1, 1, 0, 0, 1, 0, 1, 0, 1, 0]
def draw_solution(graph: nx.Graph, solution: list):
solution_nodes = [v for v in graph.nodes if solution[v]]
solution_edges = [
(u, v) for u, v in graph.edges if u in solution_nodes or v in solution_nodes
]
nx.draw_kamada_kawai(graph, with_labels=True)
nx.draw_kamada_kawai(
graph,
nodelist=solution_nodes,
edgelist=solution_edges,
node_color="r",
edge_color="y",
)
draw_solution(graph, best_solution)
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(mvc_model)
classical_solution = [int(pyo.value(mvc_model.x[i])) for i in graph.nodes]
classical_solution
[1, 1, 0, 1, 1, 0, 0, 0, 1, 0]
draw_solution(graph, classical_solution)
References
[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.