Minimum Dominating Set (MDS) Problem
The Minimum Dominating Set problem [1] is a classical NP-hard problem in computer science and graph theory. In this problem, we are given a graph, and we aim to find the smallest subset of vertices such that every node in the graph is either in the subset or is a neighbor of a node in the subset.
We represent the problem as a binary optimization problem.
Variables:
- \(x_i\) binary variables that represent whether a node \(i\) is in the dominating set or not.
Constraints:
- Every node \(i\) is either in the dominating set or connected to a node in the dominating set:
\(\forall i \in V: x_i + \sum_{j \in N(i)} x_j \geq 1\)
Where \(N(i)\) represents the neighbors of node \(i\).
Objective
- Minimize the size of the dominating set:
\(\sum_{i\in V}x_i\)
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 mds(graph: nx.Graph) -> pyo.ConcreteModel:
model = pyo.ConcreteModel()
model.x = pyo.Var(graph.nodes, domain=pyo.Binary)
@model.Constraint(graph.nodes)
def dominating_rule(model, idx):
sum_of_neighbors = sum(model.x[neighbor] for neighbor in graph.neighbors(idx))
return model.x[idx] + sum_of_neighbors >= 1
model.cost = pyo.Objective(expr=sum(model.x.values()), 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 that node is chosen for the set.
-
Constraint rule – for each node, it must be a part of the chosen set or be neighbored by one.
-
Objective rule – the sum of the variables equals the set size.
# generate a random graph
G = nx.erdos_renyi_graph(n=6, p=0.6, seed=8)
nx.draw_kamada_kawai(G, with_labels=True)
mds_model = mds(G)
mds_model.pprint()
2 Set Declarations
dominating_rule_index : Size=1, Index=None, Ordered=False
Key : Dimen : Domain : Size : Members
None : 1 : Any : 6 : {0, 1, 2, 3, 4, 5}
x_index : Size=1, Index=None, Ordered=False
Key : Dimen : Domain : Size : Members
None : 1 : Any : 6 : {0, 1, 2, 3, 4, 5}
1 Var Declarations
x : Size=6, 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
1 Objective Declarations
cost : Size=1, Index=None, Active=True
Key : Active : Sense : Expression
None : True : minimize : x[0] + x[1] + x[2] + x[3] + x[4] + x[5]
1 Constraint Declarations
dominating_rule : Size=6, Index=dominating_rule_index, Active=True
Key : Lower : Body : Upper : Active
0 : 1.0 : x[1] + x[3] + x[5] + x[0] : +Inf : True
1 : 1.0 : x[0] + x[2] + x[4] + x[1] : +Inf : True
2 : 1.0 : x[1] + x[3] + x[4] + x[5] + x[2] : +Inf : True
3 : 1.0 : x[0] + x[2] + x[4] + x[3] : +Inf : True
4 : 1.0 : x[1] + x[2] + x[3] + x[5] + x[4] : +Inf : True
5 : 1.0 : x[0] + x[2] + x[4] + x[5] : +Inf : True
5 Declarations: x_index x dominating_rule_index dominating_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 construct_combinatorial_optimization_model
from classiq.applications.combinatorial_optimization import OptimizerConfig, QAOAConfig
qaoa_config = QAOAConfig(num_layers=6, penalty_energy=8)
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=30, 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=mds_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, "minimum_dominating_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:
from classiq import show, synthesize
qprog = synthesize(qmod)
show(qprog)
Opening: https://platform.classiq.io/circuit/d5cb21cc-5e64-4e4e-8ea7-414ea1afd2ee?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(
mds_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 | |
|---|---|---|---|---|
| 707 | 0.001 | 2.0 | [1, 1, 0, 0, 0, 0] | 1 |
| 482 | 0.001 | 3.0 | [0, 0, 1, 1, 0, 1] | 1 |
| 158 | 0.001 | 3.0 | [1, 1, 0, 1, 0, 0] | 1 |
| 121 | 0.001 | 4.0 | [0, 0, 1, 1, 1, 1] | 1 |
| 704 | 0.001 | 4.0 | [1, 0, 0, 1, 1, 1] | 1 |
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()]
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(G, best_solution)
Lastly, we can compare to the classical solution of the problem:
from pyomo.opt import SolverFactory
solver = SolverFactory("couenne")
solver.solve(mds_model)
mds_model.display()
classical_solution = [int(pyo.value(mds_model.x[i])) for i in G.nodes]
Model unknown
Variables:
x : Size=6, Index=x_index
Key : Lower : Value : Upper : Fixed : Stale : Domain
0 : 0 : 1.0 : 1 : False : False : Binary
1 : 0 : 0.0 : 1 : False : False : Binary
2 : 0 : 0.0 : 1 : False : False : Binary
3 : 0 : 1.0 : 1 : False : False : Binary
4 : 0 : 0.0 : 1 : False : False : Binary
5 : 0 : 0.0 : 1 : False : False : Binary
Objectives:
cost : Size=1, Index=None, Active=True
Key : Active : Value
None : True : 2.0
Constraints:
dominating_rule : Size=6
Key : Lower : Body : Upper
0 : 1.0 : 2.0 : None
1 : 1.0 : 1.0 : None
2 : 1.0 : 1.0 : None
3 : 1.0 : 2.0 : None
4 : 1.0 : 1.0 : None
5 : 1.0 : 1.0 : None
draw_solution(G, classical_solution)
References
[1]: Dominating 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.