bundles / scipy 2.0.0.dev0+git20260907.41eeb59 / scipy / optimize / _qap / quadratic_assignment

function

`scipy.optimize._qap:quadratic_assignment`

source: build-install/usr/lib/python3.14/site-packages/scipy/optimize/_qap.py:11

Signature

def quadratic_assignment(A, B, method = faq, options = None)

Summary

Approximates solution to the quadratic assignment problem and the graph matching problem.

Extended Summary

Quadratic assignment solves problems of the following form:

\min_P & \ {\ \text{trace}(A^T P B P^T)}\\
\mbox{s.t. } & {P \ \epsilon \ \mathcal{P}}\\

where $P$ is the set of all permutation matrices, and $A$ and $B$ are square matrices.

Graph matching tries to maximize the same objective function. This algorithm can be thought of as finding the alignment of the nodes of two graphs that minimizes the number of induced edge disagreements, or, in the case of weighted graphs, the sum of squared edge weight differences.

Note that the quadratic assignment problem is NP-hard. The results given here are approximations and are not guaranteed to be optimal.

Parameters

A : 2-D array, square

The square matrix $A$ in the objective function above.

B : 2-D array, square

The square matrix $B$ in the objective function above.

method : str in {'faq', '2opt'} (default: 'faq')

The algorithm used to solve the problem. 'faq' (default) and '2opt' are available.

options : dict, optional

A dictionary of solver options. All solvers support the following:

maximize

Maximizes the objective function if True.

partial_match

Fixes part of the matching. Also known as a "seed" ^[2].

Each row of partial_match specifies a pair of matched nodes: node partial_match[i, 0] of A is matched to node partial_match[i, 1] of B. The array has shape (m, 2), where m is not greater than the number of nodes, $n$ .

rng

Pseudorandom number generator state. When rng is None, a new numpy.random.Generator is created using entropy from the operating system. Types other than numpy.random.Generator are passed to numpy.random.default_rng to instantiate a Generator.

For method-specific options, see show_options('quadratic_assignment').

Returns

res : OptimizeResult

OptimizeResult containing the following fields.

col_ind: Column indices corresponding to the best permutation found of the nodes of B.
fun: The objective value of the solution.
nit: The number of iterations performed during optimization.

Notes

The default method 'faq' uses the Fast Approximate QAP algorithm ^[1]; it typically offers the best combination of speed and accuracy. Method '2opt' can be computationally expensive, but may be a useful alternative, or it can be used to refine the solution returned by another method.

Examples

import numpy as np
from scipy.optimize import quadratic_assignment
rng = np.random.default_rng()
A = np.array([[0, 80, 150, 170], [80, 0, 130, 100],
              [150, 130, 0, 120], [170, 100, 120, 0]])
B = np.array([[0, 5, 2, 7], [0, 0, 3, 8],
              [0, 0, 0, 3], [0, 0, 0, 0]])
res = quadratic_assignment(A, B, options={'rng': rng})
print(res)

✓

     fun: 3260
 col_ind: [0 3 2 1]
     nit: 9

The see the relationship between the returned ``col_ind`` and ``fun``, use ``col_ind`` to form the best permutation matrix found, then evaluate the objective function :math:`f(P) = trace(A^T P B P^T )`.

perm = res['col_ind']
P = np.eye(len(A), dtype=int)[perm]
fun = np.trace(A.T @ P @ B @ P.T)
print(fun)

✓

Alternatively, to avoid constructing the permutation matrix explicitly, directly permute the rows and columns of the distance matrix.

fun = np.trace(A.T @ B[perm][:, perm])
print(fun)

✓

Although not guaranteed in general, ``quadratic_assignment`` happens to have found the globally optimal solution.

from itertools import permutations
perm_opt, fun_opt = None, np.inf
for perm in permutations([0, 1, 2, 3]):
    perm = np.array(perm)
    fun = np.trace(A.T @ B[perm][:, perm])
    if fun < fun_opt:
        fun_opt, perm_opt = fun, perm
print(np.array_equal(perm_opt, res['col_ind']))

✓

True

Here is an example for which the default method, :ref:`'faq' <optimize.qap-faq>`, does not find the global optimum.

A = np.array([[0, 5, 8, 6], [5, 0, 5, 1],
              [8, 5, 0, 2], [6, 1, 2, 0]])
B = np.array([[0, 1, 8, 4], [1, 0, 5, 2],
              [8, 5, 0, 5], [4, 2, 5, 0]])
res = quadratic_assignment(A, B, options={'rng': rng})
print(res)

✓

     fun: 178
 col_ind: [1 0 3 2]
     nit: 13

If accuracy is important, consider using :ref:`'2opt' <optimize.qap-2opt>` to refine the solution.

guess = np.array([np.arange(len(A)), res.col_ind]).T
res = quadratic_assignment(A, B, method="2opt",
    options = {'rng': rng, 'partial_guess': guess})
print(res)

✓

     fun: 176
 col_ind: [1 2 3 0]
     nit: 17

Aliases

scipy.optimize.quadratic_assignment