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Module « scipy.optimize »

Fonction quadratic_assignment - module scipy.optimize

Signature de la fonction quadratic_assignment

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

Description

help(scipy.optimize.quadratic_assignment)

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

Quadratic assignment solves problems of the following form:

.. math::

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

where :math:`\mathcal{P}` is the set of all permutation matrices,
and :math:`A` and :math:`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 :math:`A` in the objective function above.

B : 2-D array, square
    The square matrix :math:`B` in the objective function above.

method :  str in {'faq', '2opt'} (default: 'faq')
    The algorithm used to solve the problem.
    :ref:`'faq' <optimize.qap-faq>` (default) and
    :ref:`'2opt' <optimize.qap-2opt>` are available.

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

    maximize : bool (default: False)
        Maximizes the objective function if ``True``.

    partial_match : 2-D array of integers, optional (default: None)
        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, :math:`n`.

    rng : `numpy.random.Generator`, optional
        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``.

        .. versionchanged:: 1.15.0
            As part of the `SPEC-007 <https://scientific-python.org/specs/spec-0007/>`_
            transition from use of `numpy.random.RandomState` to
            `numpy.random.Generator` is occurring. Supplying
            `np.random.RandomState` to this function will now emit a
            `DeprecationWarning`. In SciPy 1.17 its use will raise an exception.
            In addition relying on global state using `np.random.seed`
            will emit a `FutureWarning`. In SciPy 1.17 the global random number
            generator will no longer be used.
            Use of an int-like seed will raise a `FutureWarning`, in SciPy 1.17 it
            will be normalized via `np.random.default_rng` rather than
            `np.random.RandomState`.

    For method-specific options, see
    :func:`show_options('quadratic_assignment') <show_options>`.

Returns
-------
res : OptimizeResult
    `OptimizeResult` containing the following fields.

    col_ind : 1-D array
        Column indices corresponding to the best permutation found of the
        nodes of `B`.
    fun : float
        The objective value of the solution.
    nit : int
        The number of iterations performed during optimization.

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

References
----------
.. [1] J.T. Vogelstein, J.M. Conroy, V. Lyzinski, L.J. Podrazik,
       S.G. Kratzer, E.T. Harley, D.E. Fishkind, R.J. Vogelstein, and
       C.E. Priebe, "Fast approximate quadratic programming for graph
       matching," PLOS one, vol. 10, no. 4, p. e0121002, 2015,
       :doi:`10.1371/journal.pone.0121002`

.. [2] D. Fishkind, S. Adali, H. Patsolic, L. Meng, D. Singh, V. Lyzinski,
       C. Priebe, "Seeded graph matching", Pattern Recognit. 87 (2019):
       203-215, :doi:`10.1016/j.patcog.2018.09.014`

.. [3] "2-opt," Wikipedia.
       https://en.wikipedia.org/wiki/2-opt

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)
3260

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)
3260

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

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