Set Cover Problem
Introduction
The set cover problem [1] represents a well-known problem in the fields of combinatorics, computer science, and complexity theory. It is an NP-complete problems.
The problem presents us with a universal set, \(\displaystyle U\), and a collection \(\displaystyle S\) of subsets of \(\displaystyle U\). The goal is to find the smallest possible subfamily, \(\displaystyle C \subseteq S\), whose union equals the universal set.
Formally, let's consider a universal set \(\displaystyle U = {1, 2, ..., n}\) and a collection \(\displaystyle S\) containing \(m\) subsets of \(\displaystyle U\), \(\displaystyle S = {S_1, ..., S_m}\) with \(\displaystyle S_i \subseteq U\). The challenge of the set cover problem is to find a subset \(\displaystyle C\) of \(\displaystyle S\) of minimal size such that \(\displaystyle \bigcup_{S_i \in C} S_i = U\).
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 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 itertools
from typing import List
def set_cover(sub_sets: List[List[int]]) -> pyo.ConcreteModel:
entire_set = set(itertools.chain(*sub_sets))
n = max(entire_set)
num_sets = len(sub_sets)
assert entire_set == set(
range(1, n + 1)
), f"the union of the subsets is {entire_set} not equal to range(1, {n + 1})"
model = pyo.ConcreteModel()
model.x = pyo.Var(range(num_sets), domain=pyo.Binary)
@model.Constraint(entire_set)
def independent_rule(model, num):
return sum(model.x[idx] for idx in range(num_sets) if num in sub_sets[idx]) >= 1
model.cost = pyo.Objective(expr=sum(model.x.values()), sense=pyo.minimize)
return model
The model contains:
-
Binary variable for each subset (model.x) indicating if it is included in the sub-collection.
-
Objective rule – the size of the sub-collection.
-
Constraint – the sub-collection covers the original set.
sub_sets = sub_sets = [
[1, 2, 3, 4],
[2, 3, 4, 5],
[6, 7],
[8, 9, 10],
[1, 6, 8],
[3, 7, 9],
[4, 7, 10],
[2, 5, 8],
]
set_cover_model = set_cover(sub_sets)
set_cover_model.pprint()
2 Set Declarations
independent_rule_index : Size=1, Index=None, Ordered=False
Key : Dimen : Domain : Size : Members
None : 1 : Any : 10 : {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
x_index : Size=1, Index=None, Ordered=Insertion
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 : minimize : x[0] + x[1] + x[2] + x[3] + x[4] + x[5] + x[6] + x[7]
1 Constraint Declarations
independent_rule : Size=10, Index=independent_rule_index, Active=True
Key : Lower : Body : Upper : Active
1 : 1.0 : x[0] + x[4] : +Inf : True
2 : 1.0 : x[0] + x[1] + x[7] : +Inf : True
3 : 1.0 : x[0] + x[1] + x[5] : +Inf : True
4 : 1.0 : x[0] + x[1] + x[6] : +Inf : True
5 : 1.0 : x[1] + x[7] : +Inf : True
6 : 1.0 : x[2] + x[4] : +Inf : True
7 : 1.0 : x[2] + x[5] + x[6] : +Inf : True
8 : 1.0 : x[3] + x[4] + x[7] : +Inf : True
9 : 1.0 : x[3] + x[5] : +Inf : True
10 : 1.0 : x[3] + x[6] : +Inf : True
5 Declarations: x_index x independent_rule_index independent_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 *
from classiq.applications.combinatorial_optimization import OptimizerConfig, QAOAConfig
qaoa_config = QAOAConfig(num_layers=3, penalty_energy=10)
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=set_cover_model,
qaoa_config=qaoa_config,
optimizer_config=optimizer_config,
)
We also set the quantum backend we want to execute on:
from classiq.execution import ClassiqBackendPreferences
qmod = set_execution_preferences(
qmod, backend_preferences=ClassiqBackendPreferences(backend_name="simulator")
)
write_qmod(qmod, "set_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:
qprog = synthesize(qmod)
show(qprog)
Opening: https://platform.classiq.io/circuit/11fd03ea-49d7-4ea9-a772-fca49f53953e?version=0.41.0.dev39%2B79c8fd0855
We now solve the problem by calling the execute function on the quantum program we have generated:
result = execute(qprog).result_value()
We can check the convergence of the run:
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(
set_cover_model, vqe_result=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 | |
|---|---|---|---|---|
| 118 | 0.001 | 4.0 | [0, 0, 0, 0, 1, 1, 1, 1] | 1 |
| 986 | 0.001 | 14.0 | [0, 0, 0, 0, 1, 1, 1, 1] | 1 |
| 524 | 0.001 | 14.0 | [0, 0, 0, 0, 1, 1, 1, 1] | 1 |
| 142 | 0.001 | 15.0 | [0, 0, 0, 1, 1, 1, 1, 1] | 1 |
| 37 | 0.001 | 15.0 | [1, 1, 1, 1, 1, 0, 0, 0] | 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()]
print(
f"Quantum Solution: num_sets={int(sum(best_solution))}, sets={[sub_sets[i] for i in range(len(best_solution)) if best_solution[i]]}"
)
Quantum Solution: num_sets=4, sets=[[1, 6, 8], [3, 7, 9], [4, 7, 10], [2, 5, 8]]
Lastly, we can compare to the classical solution of the problem:
from pyomo.opt import SolverFactory
solver = SolverFactory("couenne")
solver.solve(set_cover_model)
set_cover_model.display()
Model unknown
Variables:
x : Size=8, Index=x_index
Key : Lower : Value : Upper : Fixed : Stale : Domain
0 : 0 : 1.0 : 1 : False : False : Binary
1 : 0 : 1.0 : 1 : False : False : Binary
2 : 0 : 1.0 : 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
6 : 0 : 0.0 : 1 : False : False : Binary
7 : 0 : 0.0 : 1 : False : False : Binary
Objectives:
cost : Size=1, Index=None, Active=True
Key : Active : Value
None : True : 4.0
Constraints:
independent_rule : Size=10
Key : Lower : Body : Upper
1 : 1.0 : 1.0 : None
2 : 1.0 : 2.0 : None
3 : 1.0 : 2.0 : None
4 : 1.0 : 2.0 : None
5 : 1.0 : 1.0 : None
6 : 1.0 : 1.0 : None
7 : 1.0 : 1.0 : None
8 : 1.0 : 1.0 : None
9 : 1.0 : 1.0 : None
10 : 1.0 : 1.0 : None
classical_solution = [
pyo.value(set_cover_model.x[i]) for i in range(len(set_cover_model.x))
]
print(
f"Classical Solution: num_sets={int(sum(classical_solution))}, sets={[sub_sets[i] for i in range(len(classical_solution)) if classical_solution[i]]}"
)
Classical Solution: num_sets=4, sets=[[1, 2, 3, 4], [2, 3, 4, 5], [6, 7], [8, 9, 10]]
References
[1]: Set Cover Problem (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.