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.
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 itertools
from typing import List
import pyomo.core as pyo
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()
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, 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 import set_execution_preferences
from classiq.execution import ClassiqBackendPreferences, ExecutionPreferences
backend_preferences = ExecutionPreferences(
backend_preferences=ClassiqBackendPreferences(backend_name="aer_simulator")
)
qmod = set_execution_preferences(qmod, backend_preferences)
with open("set_cover.qmod", "w") as f:
f.write(qmod)
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)
We now solve the problem using the generated circuit by using the execute method:
from classiq import execute
res = execute(qprog).result()
We can check the convergence of the run:
from classiq.execution import VQESolverResult
vqe_result = res[1].value
vqe_result.convergence_graph
Optimization Results¶
We can also examine the statistics of the algorithm:
import pandas as pd
optimization_result = pd.DataFrame.from_records(res[0].value)
optimization_result.sort_values(by="cost", ascending=True).head(5)
And the histogram:
optimization_result.hist("cost", weights=optimization_result["probability"])
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]]}"
)
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()
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]]}"
)
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.