# Quadratic Assignment on Dirac

#### Device: Dirac-1

## Introduction

The quadratic assignment problem was first introduced in 1957 by Koopmans and Beckmann to solve facility location problems that call for minimizing the cost proportional to the flow of goods between the facilities. This is a well-studied problem.

QAP seeks to minimize a cost function based on fixed quantities (weights) between pairs of one set, $P$, and distances between pairs of another set $L$ when each member of $L$ is assigned to one member of $P$.

## Importance

The problem itself is based on a simplified hypothetical situation of a realistic planning problem. Although the context of the problem itself is quite different, it has the same two-way one-hot constraint structure as the traveling salesperson problem. This constraint structure comes from the fact that only one of each type of facility will be constructed, and each location can only accommodate one facility. As we will show in our example, the problem statement is more involved than with the traveling salesperson problem. It is defined by distances between locations, flows between facilities, and the cost to construct at each location. The component of the objective function leads to analogous terms to the traveling salesperson problem, such as terms that reference the location of two facilities. This is similar to how the distances that define traveling salesperson problem costs reference the times at which two cities are visited. The cost of building a facility at each location, however, provides terms that reference only the location of a single facility. This extra cost means that quadratic assignment has less symmetry than the traveling salesperson problem. While the cost is the same in the traveling salesperson problem regardless of the starting city (assuming the salesperson must end in that city), there is no analogous symmetry for quadratic assignment.

## Applications

As with many other problems, the quadratic assignment problem is motivated by a real task, deciding where to construct industrial facilities. While a real situation may involve additional constraints (such as regulations that do not allow a factory that works with dangerous chemicals to be located in certain parts of a city), this problem presents a simplified framework for understanding planning problems. In addition to the obvious planning application, which is how the problem is formulated, there are other less obvious applications as reviewed in this paper. One such example is hospital design, where the flow is that of patients rather than materials. Similarly, the problem of placing components on a computer backboard to minimize the length of wires can be phrased as a QAP. In this example, the wires are the flow, and the components are the facilities, representing an empty location as a facility with no cost and no flows. An even more exotic application of this problem is forest management, where facilities are representations of zoning of a location for specific use. In this case, there will be multiple facilities of the same type, and the flows represent the desirability (or lack thereof) of having different types of areas in proximity to each other. For example, having an area zoned as a wildlife preserve next to an area zoned for recreation is desirable, while having recreation next to logging is not.

## Simple Example

This first example is a synthetic problem with five facilities and five locations. The first consideration is the planned flow between the facilities. The table below shows the quantity of material planned to be moved from one facility to the other. Facility 1 sends to facilities 2, 3, and 5. Facilities 2 and 3 send to facilities 4 and 5. Facility 4 doesn't send to any other facility, and facility 5 sends one unit to facility 4.

Source Facility / Destination Facility | Facility 1 | Facility 2 | Faclity 3 | Facility 4 | Facility 5 |
---|---|---|---|---|---|

Facility 1 | 5 | 8 | 1 | ||

Facility 2 | 10 | 15 | |||

Facility 3 | 13 | 18 | |||

Facility 4 | |||||

Facility 5 | 1 |

The second consideration is the distance between locations. The distances in this example are symmetric.

Distances | Location 1 | Location 2 | Location 3 | Location 4 | Location 5 |
---|---|---|---|---|---|

Location 1 | 0 | 8.54 | 6.4 | 10 | 8.94 |

Location 2 | 8.54 | 0 | 4.47 | 5.39 | 6.49 |

Location 3 | 6.4 | 4.47 | 0 | 3.61 | 3 |

Location 4 | 10 | 5.39 | 3.61 | 0 | 2 |

Location 5 | 8.94 | 6.49 | 3 | 2 | 0 |

The third consideration is the cost of building a facility at a location.

Facility / Location | Location 1 | Location 2 | Location 3 | Location 4 | Location 5 |
---|---|---|---|---|---|

Facility 1 | 2 | 3 | 6 | 3 | 7 |

Facility 2 | 3 | 9 | 2 | 5 | 9 |

Facility 3 | 2 | 6 | 4 | 1 | 2 |

Facility 4 | 7 | 5 | 8 | 5 | 7 |

Facility 5 | 1 | 9 | 2 | 9 | 2 |

The outcomes in this example range from optimal to 166% of optimal. This implies a considerable benefit to performing the optimization.

In [1]:

- import numpy as np
- import matplotlib.pyplot as plt
- from qci_client import QciClient
- from helpers import plot_qap, assignment_from_solution, create_qap_objective, create_qap_constraints, find_index_of_nearest
- from qap_data import mip_obj_vals

Here is the data described in the tables above.

In [2]:

- A = np.array([[0, 5, 8, 0, 1],
- [0, 0, 0, 10, 15],
- [0, 0, 0, 13, 18],
- [0, 0, 0, 0, 0.],
- [0, 0, 0, 1, 0.]])
- B = np.array([[0, 8.54, 6.4, 10, 8.94],
- [8.54, 0, 4.47, 5.39, 6.49],
- [6.4, 4.47, 0, 3.61, 3.0],
- [10, 5.39, 3.61, 0, 2.0],
- [8.94, 6.49, 3.0, 2.0, 0.]])
- C = np.array([[2, 3, 6, 3, 7],
- [3, 9, 2, 5, 9],
- [2, 6, 4, 1, 2],
- [7, 5, 8, 5, 7],
- [1, 9, 2, 9, 2.]])
- n = 5
- num_variables = 25

This code creates the objective matrix from the three input matrices, and then uploads the data to the API.

In [3]:

- token = "your_token"
- api_url = "https://api.qci-prod.com"
- qciclient = QciClient(api_token=token, url=api_url)

Here we start a client for the API and upload the objective matrix.

The constraint matrix is built using only the number of facilities/locations. It is the same across all problems of the same size, independent of the other input data.

This code builds the constraint data and uploads two files, one for the left-hand side and another for the right.

In [4]:

- constraint_file = create_qap_constraints(n)
- constraint_file_id = qciclient.upload_file(file=constraint_file)["file_id"]

In [5]:

- objective_file = create_qap_objective(A, B, C, n, num_variables)
- obj_file_id = qciclient.upload_file(file=objective_file)["file_id"]

This is where we make the job request. The request is synchronous, but could be made asynchronously and results retrieved at a later time.

In [6]:

- alpha = 105.625
- job_params = {"device_type": "dirac-1", "alpha": alpha, "num_samples": 5}
- body = qciclient.build_job_body(job_type="sample-constraint", job_params=job_params,
- constraints_file_id=constraint_file_id, objective_file_id=obj_file_id,
- job_name=f"QAP Demo",
- job_tags=[])
- # submit the job request to be processed asynchronously
- response = qciclient.process_job(job_body=body)

Out [ ]:

2024-05-08 18:30:31 - Dirac allocation balance = 0 s (unmetered) 2024-05-08 18:30:31 - Job submitted: job_id='663c2737d448b017e54f94d2' 2024-05-08 18:30:31 - QUEUED 2024-05-08 18:33:26 - RUNNING 2024-05-08 18:48:42 - COMPLETED 2024-05-08 18:48:45 - Dirac allocation balance = 0 s (unmetered)

The response includes results samples and energies.

In [7]:

- response['results']

Out [7]:

{'counts': [3, 2], 'energies': [-725.6101734638214, -724.4597828388214], 'feasibilities': [True, False], 'objective_values': [330.6400022506714, 120.53999900817871], 'solutions': [[1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1], [0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1]]}

In [8]:

- results = response["results"]
- sample = np.array(results["solutions"])

We convert the bit vector of length 25 into an array of assignments of length 5 and plot the assignments in a bi-partite graph.

In [9]:

- assignment = assignment_from_solution(sample[0], n)
- assignment

Out [9]:

[0, 3, 2, 1, 4]

In [10]:

- plot_qap(assignment)

Out [ ]:

<Figure size 640x480 with 1 Axes>

These values were obtained by enumerating all feasible solutions and evaluating the objective function.

## Results

In this plot, the objective values of all feasible solutions are shown with the objective value of the solution found by **Dirac 1**. Notice the result is the lowest possible objective value.

In [11]:

- plt.figure(figsize=(12, 9))
- ser1 = plt.plot(mip_obj_vals, "c-", label="Feasible Solutions")
- plt.title("Feasible solutions to QAP")
- ax = plt.gca()
- plt.xlabel("Feasible Solution")
- ax.get_xaxis().set_visible(False)
- plt.ylabel("Objective Value")
- # ser2 = plt.plot([0, len(mip_obj_vals)-1], [results["energies"][0], results["energies"][0]], label="Found Solution")
- obj_val = results["objective_values"][0]
- idx = find_index_of_nearest(mip_obj_vals, obj_val)
- ser2 = plt.scatter([idx], [obj_val], c="r", marker="o", label="Found Solution")
- plt.legend()
- plt.grid()

Out [ ]:

<Figure size 1200x900 with 1 Axes>

## Conclusion

In this tutorial, we have learned about the quadratic assignment problem. While distinct, quadratic assignment shares the same constraint structure as the traveling salesperson problem. While less well known than the traveling salesperson problem, quadratic assignment is a rich problem that has many practical applications. A logical next step could be to explore other constrained problems that our devices can solve through the quadratic linearly constrained binary optimization page. Of course, another option is to start using our device to solve some of your own optimization problems.