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Subgraph Isomorphism Example

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The Subgraph Isomorphism problem determines whether a smaller pattern graph can be found as a subgraph of a larger target graph, preserving edges. It is NP-complete and used in pattern recognition and database querying.

import getpass
import os

import numpy as np
from dotenv import load_dotenv
from luna_quantum.algorithms import SCIP

from luna_usecases.subgraph_isomorphism import (
    SubgraphIsomorphismCollection,
    SubgraphIsomorphismData,
    SubgraphIsomorphismFormulation,
    SubgraphIsomorphismInstance,
)

load_dotenv()
if "LUNA_API_KEY" not in os.environ:
    os.environ["LUNA_API_KEY"] = getpass.getpass("Enter your Luna API key: ")

Create Data

Define a 3-node triangle pattern and a 5-node target graph containing it.

adj_pattern = np.array(
    [
        [0, 1, 1],
        [1, 0, 1],
        [1, 1, 0],
    ]
)
adj_target = np.array(
    [
        [0, 1, 1, 0, 0],
        [1, 0, 1, 1, 0],
        [1, 1, 0, 0, 1],
        [0, 1, 0, 0, 1],
        [0, 0, 1, 1, 0],
    ]
)
node_names_pattern = ["P1", "P2", "P3"]
node_names_target = ["A", "B", "C", "D", "E"]
data = SubgraphIsomorphismData.from_adjacency_matrices(
    adjacency_matrix_pattern=adj_pattern,
    adjacency_matrix_target=adj_target,
    node_names_pattern=node_names_pattern,
    node_names_target=node_names_target,
)
print(data.to_string())

Subgraph Isomorphism Data:
  Pattern: 3 nodes, 3 edges
  Target: 5 nodes, 6 edges

Plot Data

Visualize the pattern and target graphs.

data.plot()


(<Axes: title={'center': 'Pattern'}>, <Axes: title={'center': 'Target'}>)
png

Create Formulation

Find a mapping of pattern nodes into target nodes that preserves all edges.

formulation = SubgraphIsomorphismFormulation()
print(formulation.to_string(data))

Subgraph Isomorphism Formulation (undirected graphs):
  Pattern nodes: 3, Target nodes: 5

Index:
  i, k -- pattern node indices with i < k, i = 0, ..., 2
  j, m -- target node indices with j < m, j = 0, ..., 4

Decision Variables:
  x[i,j] in {0,1} for all i, j
  x[i,j] = 1 if pattern node i maps to target node j
  Total: 15 binary variables

Objective:
  None (feasibility problem)

Constraints:
  1. Each pattern node maps to exactly one target node (3 constraints):
     sum_j x[i,j] == 1  for all i = 0, ..., 2
  2. Each target node mapped from at most one pattern node (5 constraints):
     sum_i x[i,j] <= 1  for all j = 0, ..., 4
  3. Edge preservation (12 constraints, undirected pairs):
     x[i,j] + x[k,m] <= 1
     for edge (i,k) in pattern with i < k
     and non-edge (j,m) in target with j < m

Create Instance

Combine data and formulation into a solvable instance.

instance = SubgraphIsomorphismInstance(data=data, formulation=formulation)
print(instance.to_string())

Data:Subgraph Isomorphism Data:
  Pattern: 3 nodes, 3 edges
  Target: 5 nodes, 6 edges
Formulation:Subgraph Isomorphism Formulation (undirected graphs):
  Pattern nodes: 3, Target nodes: 5

Index:
  i, k -- pattern node indices with i < k, i = 0, ..., 2
  j, m -- target node indices with j < m, j = 0, ..., 4

Decision Variables:
  x[i,j] in {0,1} for all i, j
  x[i,j] = 1 if pattern node i maps to target node j
  Total: 15 binary variables

Objective:
  None (feasibility problem)

Constraints:
  1. Each pattern node maps to exactly one target node (3 constraints):
     sum_j x[i,j] == 1  for all i = 0, ..., 2
  2. Each target node mapped from at most one pattern node (5 constraints):
     sum_i x[i,j] <= 1  for all j = 0, ..., 4
  3. Edge preservation (12 constraints, undirected pairs):
     x[i,j] + x[k,m] <= 1
     for edge (i,k) in pattern with i < k
     and non-edge (j,m) in target with j < m

Formulate Model

Translate the instance into a mathematical optimization model.

model = instance.formulate()

Solve and Interpret

Solve the model with SCIP and interpret the raw result into a use-case-specific solution.

scip = SCIP()
job = scip.run(model)
sol = job.result()
uc_solution = instance.interpret(sol)
print(uc_solution.to_string())

/Users/maximilianjanetschek/PycharmProjects/luna-usecases/.venv/lib/python3.13/site-packages/rich/live.py:260:
UserWarning: install "ipywidgets" for Jupyter support
  warnings.warn('install "ipywidgets" for Jupyter support')










2026-05-29 11:35:57 INFO     Sleeping for 5.0 seconds. Waiting and checking a function in a loop.










Subgraph Isomorphism Solution:
  Valid: True
  Mapping: {'P1': 'A', 'P2': 'B', 'P3': 'C'}



Plot Solution


Visualize the optimal solution.


uc_solution.plot(data)



<Axes: title={'center': 'Subgraph Isomorphism — valid=True'}>


png


Collections


Generate a benchmark collection of random instances for batch processing.


collection = SubgraphIsomorphismCollection.from_random(
    min_pattern=3,
    max_pattern=5,
    num_instances=2,
    seed=42,
)
model = collection.instances[0].formulate()
print(model)



Model: subgraph_isomorphism<subgraph_isomorphism>
Minimize

Subject To
  pattern_node_0: x_0_0 + x_0_1 + x_0_2 + x_0_3 + x_0_4 == 1
  pattern_node_1: x_1_0 + x_1_1 + x_1_2 + x_1_3 + x_1_4 == 1
  pattern_node_2: x_2_0 + x_2_1 + x_2_2 + x_2_3 + x_2_4 == 1
  target_node_0: x_0_0 + x_1_0 + x_2_0 <= 1
  target_node_1: x_0_1 + x_1_1 + x_2_1 <= 1
  target_node_2: x_0_2 + x_1_2 + x_2_2 <= 1
  target_node_3: x_0_3 + x_1_3 + x_2_3 <= 1
  target_node_4: x_0_4 + x_1_4 + x_2_4 <= 1
  edge_pres_0_1_0_3: x_0_0 + x_1_3 <= 1
  edge_pres_0_1_0_4: x_0_0 + x_1_4 <= 1
  edge_pres_0_1_2_3: x_0_2 + x_1_3 <= 1
  edge_pres_0_1_2_4: x_0_2 + x_1_4 <= 1
  edge_pres_0_1_3_4: x_0_3 + x_1_4 <= 1
  edge_pres_0_2_0_3: x_0_0 + x_2_3 <= 1
  edge_pres_0_2_0_4: x_0_0 + x_2_4 <= 1
  edge_pres_0_2_2_3: x_0_2 + x_2_3 <= 1
  edge_pres_0_2_2_4: x_0_2 + x_2_4 <= 1
  edge_pres_0_2_3_4: x_0_3 + x_2_4 <= 1
  edge_pres_1_2_0_3: x_1_0 + x_2_3 <= 1
  edge_pres_1_2_0_4: x_1_0 + x_2_4 <= 1
  edge_pres_1_2_2_3: x_1_2 + x_2_3 <= 1
  edge_pres_1_2_2_4: x_1_2 + x_2_4 <= 1
  edge_pres_1_2_3_4: x_1_3 + x_2_4 <= 1
Binary
  x_0_0 x_0_1 x_0_2 x_0_3 x_0_4 x_1_0 x_1_1 x_1_2 x_1_3 x_1_4 x_2_0 x_2_1 x_2_2
  x_2_3 x_2_4