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.
import networkx as nx
import numpy as np
import pyomo.core as pyo
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 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 CombinatorialProblem quantum object. Under the hood it tranlastes the Pyomo model to a quantum model of the QAOA algorithm, with a cost function translated from the Pyomo model. We can choose the number of layers for the QAOA ansatz using the argument num_layers, and the penalty_factor, which will be the coefficient of the constraints term in the cost hamiltonian.
from classiq import *
from classiq.applications.combinatorial_optimization import CombinatorialProblem
combi = CombinatorialProblem(pyo_model=mds_model, num_layers=6, penalty_factor=10)
qmod = combi.get_model()
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:
qprog = combi.get_qprog()
show(qprog)
Opening: https://platform.classiq.io/circuit/2uni9DhUGnElonQI7CJ3VnTl921?login=True&version=0.72.1
We now solve the problem by calling the optimize method of the CombinatorialProblem object. For the classical optimization part of the QAOA algorithm we define the maximum number of classical iterations (maxiter) and the \(\alpha\)-parameter (quantile) for running CVaR-QAOA, an improved variation of the QAOA algorithm [3]:
optimized_params = combi.optimize(maxiter=70, quantile=0.7)
Optimization Progress: 71it [09:18, 7.87s/it]
We can check the convergence of the run:
import matplotlib.pyplot as plt
fig, axes = plt.subplots(nrows=1, ncols=1)
axes.plot(combi.cost_trace)
axes.set_xlabel("Iterations")
axes.set_ylabel("Cost")
axes.set_title("Cost convergence")
Text(0.5, 1.0, 'Cost convergence')
Optimization Results
We can also examine the statistics of the algorithm. In order to get samples with the optimized parameters, we call the get_results method:
optimization_result = combi.sample(optimized_params)
optimization_result.sort_values(by="cost").head(5)
| solution | probability | cost | |
|---|---|---|---|
| 1376 | {'x': [0, 1, 0, 1, 0, 1], 'dominating_rule_0_s... | 0.000488 | 3.0 |
| 67 | {'x': [0, 1, 1, 0, 1, 0], 'dominating_rule_0_s... | 0.000488 | 3.0 |
| 1588 | {'x': [0, 1, 0, 1, 1, 0], 'dominating_rule_0_s... | 0.000488 | 3.0 |
| 301 | {'x': [0, 1, 0, 1, 1, 0], 'dominating_rule_0_s... | 0.000488 | 3.0 |
| 1293 | {'x': [1, 1, 0, 1, 0, 1], 'dominating_rule_0_s... | 0.000488 | 4.0 |
We will also want to 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)
Let us plot the solution:
best_solution = optimization_result.solution[optimization_result.cost.idxmin()]
best_solution
{'x': [0, 1, 1, 0, 1, 0],
'dominating_rule_0_slack_var': [0, 0],
'dominating_rule_1_slack_var': [0, 1],
'dominating_rule_2_slack_var': [0, 1, 0],
'dominating_rule_3_slack_var': [1, 0],
'dominating_rule_4_slack_var': [0, 1, 0],
'dominating_rule_5_slack_var': [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(G, [best_solution["x"][i] for i in range(len(best_solution["x"]))])
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.