text: import numpy as np
  import operator
  from . import (linear_sum_assignment, OptimizeResult)
  from ._optimize import _check_unknown_options
  
  from scipy._lib._util import check_random_state
  import itertools
  
  QUADRATIC_ASSIGNMENT_METHODS = ['faq', '2opt']
  
  def quadratic_assignment(A, B, method="faq", options=None):
      r"""
      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 : {None, int, `numpy.random.Generator`,
                 `numpy.random.RandomState`}, optional
  
              If `seed` is None (or `np.random`), the `numpy.random.RandomState`
              singleton is used.
              If `seed` is an int, a new ``RandomState`` instance is used,
              seeded with `seed`.
              If `seed` is already a ``Generator`` or ``RandomState`` instance then
              that instance is used.
  
          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
      --------
      >>> from scipy.optimize import quadratic_assignment
      >>> 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)
      >>> print(res)
       col_ind: array([0, 3, 2, 1])
           fun: 3260
           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)
      >>> print(res)
       col_ind: array([1, 0, 3, 2])
           fun: 178
           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 = {'partial_guess': guess})
      >>> print(res)
       col_ind: array([1, 2, 3, 0])
           fun: 176
           nit: 17
  
      """
  
      if options is None:
          options = {}
  
      method = method.lower()
      methods = {"faq": _quadratic_assignment_faq,
                 "2opt": _quadratic_assignment_2opt}
      if method not in methods:
          raise ValueError(f"method {method} must be in {methods}.")
      res = methods[method](A, B, **options)
      return res
  
  
  def _calc_score(A, B, perm):
      # equivalent to objective function but avoids matmul
      return np.sum(A * B[perm][:, perm])
  
  
  def _common_input_validation(A, B, partial_match):
      A = np.atleast_2d(A)
      B = np.atleast_2d(B)
  
      if partial_match is None:
          partial_match = np.array([[], []]).T
      partial_match = np.atleast_2d(partial_match).astype(int)
  
      msg = None
      if A.shape[0] != A.shape[1]:
          msg = "`A` must be square"
      elif B.shape[0] != B.shape[1]:
          msg = "`B` must be square"
      elif A.ndim != 2 or B.ndim != 2:
          msg = "`A` and `B` must have exactly two dimensions"
      elif A.shape != B.shape:
          msg = "`A` and `B` matrices must be of equal size"
      elif partial_match.shape[0] > A.shape[0]:
          msg = "`partial_match` can have only as many seeds as there are nodes"
      elif partial_match.shape[1] != 2:
          msg = "`partial_match` must have two columns"
      elif partial_match.ndim != 2:
          msg = "`partial_match` must have exactly two dimensions"
      elif (partial_match < 0).any():
          msg = "`partial_match` must contain only positive indices"
      elif (partial_match >= len(A)).any():
          msg = "`partial_match` entries must be less than number of nodes"
      elif (not len(set(partial_match[:, 0])) == len(partial_match[:, 0]) or
            not len(set(partial_match[:, 1])) == len(partial_match[:, 1])):
          msg = "`partial_match` column entries must be unique"
  
      if msg is not None:
          raise ValueError(msg)
  
      return A, B, partial_match
  
  
  def _quadratic_assignment_faq(A, B,
                                maximize=False, partial_match=None, rng=None,
                                P0="barycenter", shuffle_input=False, maxiter=30,
                                tol=0.03, **unknown_options):
      r"""Solve the quadratic assignment problem (approximately).
  
      This function solves the Quadratic Assignment Problem (QAP) and the
      Graph Matching Problem (GMP) using the Fast Approximate QAP Algorithm
      (FAQ) [1]_.
  
      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. This is the method-specific
          documentation for 'faq'.
          :ref:`'2opt' <optimize.qap-2opt>` is also available.
  
      Options
      -------
      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 : {None, int, `numpy.random.Generator`,
             `numpy.random.RandomState`}, optional
  
          If `seed` is None (or `np.random`), the `numpy.random.RandomState`
          singleton is used.
          If `seed` is an int, a new ``RandomState`` instance is used,
          seeded with `seed`.
          If `seed` is already a ``Generator`` or ``RandomState`` instance then
          that instance is used.
      P0 : 2-D array, "barycenter", or "randomized" (default: "barycenter")
          Initial position. Must be a doubly-stochastic matrix [3]_.
  
          If the initial position is an array, it must be a doubly stochastic
          matrix of size :math:`m' \times m'` where :math:`m' = n - m`.
  
          If ``"barycenter"`` (default), the initial position is the barycenter
          of the Birkhoff polytope (the space of doubly stochastic matrices).
          This is a :math:`m' \times m'` matrix with all entries equal to
          :math:`1 / m'`.
  
          If ``"randomized"`` the initial search position is
          :math:`P_0 = (J + K) / 2`, where :math:`J` is the barycenter and
          :math:`K` is a random doubly stochastic matrix.
      shuffle_input : bool (default: False)
          Set to `True` to resolve degenerate gradients randomly. For
          non-degenerate gradients this option has no effect.
      maxiter : int, positive (default: 30)
          Integer specifying the max number of Frank-Wolfe iterations performed.
      tol : float (default: 0.03)
          Tolerance for termination. Frank-Wolfe iteration terminates when
          :math:`\frac{||P_{i}-P_{i+1}||_F}{\sqrt{m')}} \leq tol`,
          where :math:`i` is the iteration number.
  
      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 Frank-Wolfe iterations performed.
  
      Notes
      -----
      The algorithm may be sensitive to the initial permutation matrix (or
      search "position") due to the possibility of several local minima
      within the feasible region. A barycenter initialization is more likely to
      result in a better solution than a single random initialization. However,
      calling ``quadratic_assignment`` several times with different random
      initializations may result in a better optimum at the cost of longer
      total execution time.
  
      Examples
      --------
      As mentioned above, a barycenter initialization often results in a better
      solution than a single random initialization.
  
      >>> from numpy.random import default_rng
      >>> rng = default_rng()
      >>> n = 15
      >>> A = rng.random((n, n))
      >>> B = rng.random((n, n))
      >>> res = quadratic_assignment(A, B)  # FAQ is default method
      >>> print(res.fun)
      46.871483385480545  # may vary
  
      >>> options = {"P0": "randomized"}  # use randomized initialization
      >>> res = quadratic_assignment(A, B, options=options)
      >>> print(res.fun)
      47.224831071310625 # may vary
  
      However, consider running from several randomized initializations and
      keeping the best result.
  
      >>> res = min([quadratic_assignment(A, B, options=options)
      ...            for i in range(30)], key=lambda x: x.fun)
      >>> print(res.fun)
      46.671852533681516 # may vary
  
      The '2-opt' method can be used to further refine the results.
  
      >>> options = {"partial_guess": np.array([np.arange(n), res.col_ind]).T}
      >>> res = quadratic_assignment(A, B, method="2opt", options=options)
      >>> print(res.fun)
      46.47160735721583 # may vary
  
      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] "Doubly stochastic Matrix," Wikipedia.
             https://en.wikipedia.org/wiki/Doubly_stochastic_matrix
  
      """
  
      _check_unknown_options(unknown_options)
  
      maxiter = operator.index(maxiter)
  
      # ValueError check
      A, B, partial_match = _common_input_validation(A, B, partial_match)
  
      msg = None
      if isinstance(P0, str) and P0 not in {'barycenter', 'randomized'}:
          msg = "Invalid 'P0' parameter string"
      elif maxiter <= 0:
          msg = "'maxiter' must be a positive integer"
      elif tol <= 0:
          msg = "'tol' must be a positive float"
      if msg is not None:
          raise ValueError(msg)
  
      rng = check_random_state(rng)
      n = len(A)  # number of vertices in graphs
      n_seeds = len(partial_match)  # number of seeds
      n_unseed = n - n_seeds
  
      # [1] Algorithm 1 Line 1 - choose initialization
      if not isinstance(P0, str):
          P0 = np.atleast_2d(P0)
          if P0.shape != (n_unseed, n_unseed):
              msg = "`P0` matrix must have shape m' x m', where m'=n-m"
          elif ((P0 < 0).any() or not np.allclose(np.sum(P0, axis=0), 1)
                or not np.allclose(np.sum(P0, axis=1), 1)):
              msg = "`P0` matrix must be doubly stochastic"
          if msg is not None:
              raise ValueError(msg)
      elif P0 == 'barycenter':
          P0 = np.ones((n_unseed, n_unseed)) / n_unseed
      elif P0 == 'randomized':
          J = np.ones((n_unseed, n_unseed)) / n_unseed
          # generate a nxn matrix where each entry is a random number [0, 1]
          # would use rand, but Generators don't have it
          # would use random, but old mtrand.RandomStates don't have it
          K = _doubly_stochastic(rng.uniform(size=(n_unseed, n_unseed)))
          P0 = (J + K) / 2
  
      # check trivial cases
      if n == 0 or n_seeds == n:
          score = _calc_score(A, B, partial_match[:, 1])
          res = {"col_ind": partial_match[:, 1], "fun": score, "nit": 0}
          return OptimizeResult(res)
  
      obj_func_scalar = 1
      if maximize:
          obj_func_scalar = -1
  
      nonseed_B = np.setdiff1d(range(n), partial_match[:, 1])
      if shuffle_input:
          nonseed_B = rng.permutation(nonseed_B)
  
      nonseed_A = np.setdiff1d(range(n), partial_match[:, 0])
      perm_A = np.concatenate([partial_match[:, 0], nonseed_A])
      perm_B = np.concatenate([partial_match[:, 1], nonseed_B])
  
      # definitions according to Seeded Graph Matching [2].
      A11, A12, A21, A22 = _split_matrix(A[perm_A][:, perm_A], n_seeds)
      B11, B12, B21, B22 = _split_matrix(B[perm_B][:, perm_B], n_seeds)
      const_sum = A21 @ B21.T + A12.T @ B12
  
      P = P0
      # [1] Algorithm 1 Line 2 - loop while stopping criteria not met
      for n_iter in range(1, maxiter+1):
          # [1] Algorithm 1 Line 3 - compute the gradient of f(P) = -tr(APB^tP^t)
          grad_fp = (const_sum + A22 @ P @ B22.T + A22.T @ P @ B22)
          # [1] Algorithm 1 Line 4 - get direction Q by solving Eq. 8
          _, cols = linear_sum_assignment(grad_fp, maximize=maximize)
          Q = np.eye(n_unseed)[cols]
  
          # [1] Algorithm 1 Line 5 - compute the step size
          # Noting that e.g. trace(Ax) = trace(A)*x, expand and re-collect
          # terms as ax**2 + bx + c. c does not affect location of minimum
          # and can be ignored. Also, note that trace(A@B) = (A.T*B).sum();
          # apply where possible for efficiency.
          R = P - Q
          b21 = ((R.T @ A21) * B21).sum()
          b12 = ((R.T @ A12.T) * B12.T).sum()
          AR22 = A22.T @ R
          BR22 = B22 @ R.T
          b22a = (AR22 * B22.T[cols]).sum()
          b22b = (A22 * BR22[cols]).sum()
          a = (AR22.T * BR22).sum()
          b = b21 + b12 + b22a + b22b
          # critical point of ax^2 + bx + c is at x = -d/(2*e)
          # if a * obj_func_scalar > 0, it is a minimum
          # if minimum is not in [0, 1], only endpoints need to be considered
          if a*obj_func_scalar > 0 and 0 <= -b/(2*a) <= 1:
              alpha = -b/(2*a)
          else:
              alpha = np.argmin([0, (b + a)*obj_func_scalar])
  
          # [1] Algorithm 1 Line 6 - Update P
          P_i1 = alpha * P + (1 - alpha) * Q
          if np.linalg.norm(P - P_i1) / np.sqrt(n_unseed) < tol:
              P = P_i1
              break
          P = P_i1
      # [1] Algorithm 1 Line 7 - end main loop
  
      # [1] Algorithm 1 Line 8 - project onto the set of permutation matrices
      _, col = linear_sum_assignment(P, maximize=True)
      perm = np.concatenate((np.arange(n_seeds), col + n_seeds))
  
      unshuffled_perm = np.zeros(n, dtype=int)
      unshuffled_perm[perm_A] = perm_B[perm]
  
      score = _calc_score(A, B, unshuffled_perm)
      res = {"col_ind": unshuffled_perm, "fun": score, "nit": n_iter}
      return OptimizeResult(res)
  
  
  def _split_matrix(X, n):
      # definitions according to Seeded Graph Matching [2].
      upper, lower = X[:n], X[n:]
      return upper[:, :n], upper[:, n:], lower[:, :n], lower[:, n:]
  
  
  def _doubly_stochastic(P, tol=1e-3):
      # Adapted from @btaba implementation
      # https://github.com/btaba/sinkhorn_knopp
      # of Sinkhorn-Knopp algorithm
      # https://projecteuclid.org/euclid.pjm/1102992505
  
      max_iter = 1000
      c = 1 / P.sum(axis=0)
      r = 1 / (P @ c)
      P_eps = P
  
      for it in range(max_iter):
          if ((np.abs(P_eps.sum(axis=1) - 1) < tol).all() and
                  (np.abs(P_eps.sum(axis=0) - 1) < tol).all()):
              # All column/row sums ~= 1 within threshold
              break
  
          c = 1 / (r @ P)
          r = 1 / (P @ c)
          P_eps = r[:, None] * P * c
  
      return P_eps
  
  
  def _quadratic_assignment_2opt(A, B, maximize=False, rng=None,
                                 partial_match=None,
                                 partial_guess=None,
                                 **unknown_options):
      r"""Solve the quadratic assignment problem (approximately).
  
      This function solves the Quadratic Assignment Problem (QAP) and the
      Graph Matching Problem (GMP) using the 2-opt algorithm [1]_.
  
      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. This is the method-specific
          documentation for '2opt'.
          :ref:`'faq' <optimize.qap-faq>` is also available.
  
      Options
      -------
      maximize : bool (default: False)
          Maximizes the objective function if ``True``.
      rng : {None, int, `numpy.random.Generator`,
             `numpy.random.RandomState`}, optional
  
          If `seed` is None (or `np.random`), the `numpy.random.RandomState`
          singleton is used.
          If `seed` is an int, a new ``RandomState`` instance is used,
          seeded with `seed`.
          If `seed` is already a ``Generator`` or ``RandomState`` instance then
          that instance is used.
      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`.
      partial_guess : 2-D array of integers, optional (default: None)
          A guess for the matching between the two matrices. Unlike
          `partial_match`, `partial_guess` does not fix the indices; they are
          still free to be optimized.
  
          Each row of `partial_guess` specifies a pair of matched nodes: node
          ``partial_guess[i, 0]`` of `A` is matched to node
          ``partial_guess[i, 1]`` of `B`. The array has shape ``(m, 2)``,
          where ``m`` is not greater than the number of nodes, :math:`n`.
  
      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
      -----
      This is a greedy algorithm that works similarly to bubble sort: beginning
      with an initial permutation, it iteratively swaps pairs of indices to
      improve the objective function until no such improvements are possible.
  
      References
      ----------
      .. [1] "2-opt," Wikipedia.
             https://en.wikipedia.org/wiki/2-opt
  
      .. [2] D. Fishkind, S. Adali, H. Patsolic, L. Meng, D. Singh, V. Lyzinski,
             C. Priebe, "Seeded graph matching", Pattern Recognit. 87 (2019):
             203-215, https://doi.org/10.1016/j.patcog.2018.09.014
  
      """
      _check_unknown_options(unknown_options)
      rng = check_random_state(rng)
      A, B, partial_match = _common_input_validation(A, B, partial_match)
  
      N = len(A)
      # check trivial cases
      if N == 0 or partial_match.shape[0] == N:
          score = _calc_score(A, B, partial_match[:, 1])
          res = {"col_ind": partial_match[:, 1], "fun": score, "nit": 0}
          return OptimizeResult(res)
  
      if partial_guess is None:
          partial_guess = np.array([[], []]).T
      partial_guess = np.atleast_2d(partial_guess).astype(int)
  
      msg = None
      if partial_guess.shape[0] > A.shape[0]:
          msg = ("`partial_guess` can have only as "
                 "many entries as there are nodes")
      elif partial_guess.shape[1] != 2:
          msg = "`partial_guess` must have two columns"
      elif partial_guess.ndim != 2:
          msg = "`partial_guess` must have exactly two dimensions"
      elif (partial_guess < 0).any():
          msg = "`partial_guess` must contain only positive indices"
      elif (partial_guess >= len(A)).any():
          msg = "`partial_guess` entries must be less than number of nodes"
      elif (not len(set(partial_guess[:, 0])) == len(partial_guess[:, 0]) or
            not len(set(partial_guess[:, 1])) == len(partial_guess[:, 1])):
          msg = "`partial_guess` column entries must be unique"
      if msg is not None:
          raise ValueError(msg)
  
      fixed_rows = None
      if partial_match.size or partial_guess.size:
          # use partial_match and partial_guess for initial permutation,
          # but randomly permute the rest.
          guess_rows = np.zeros(N, dtype=bool)
          guess_cols = np.zeros(N, dtype=bool)
          fixed_rows = np.zeros(N, dtype=bool)
          fixed_cols = np.zeros(N, dtype=bool)
          perm = np.zeros(N, dtype=int)
  
          rg, cg = partial_guess.T
          guess_rows[rg] = True
          guess_cols[cg] = True
          perm[guess_rows] = cg
  
          # match overrides guess
          rf, cf = partial_match.T
          fixed_rows[rf] = True
          fixed_cols[cf] = True
          perm[fixed_rows] = cf
  
          random_rows = ~fixed_rows & ~guess_rows
          random_cols = ~fixed_cols & ~guess_cols
          perm[random_rows] = rng.permutation(np.arange(N)[random_cols])
      else:
          perm = rng.permutation(np.arange(N))
  
      best_score = _calc_score(A, B, perm)
  
      i_free = np.arange(N)
      if fixed_rows is not None:
          i_free = i_free[~fixed_rows]
  
      better = operator.gt if maximize else operator.lt
      n_iter = 0
      done = False
      while not done:
          # equivalent to nested for loops i in range(N), j in range(i, N)
          for i, j in itertools.combinations_with_replacement(i_free, 2):
              n_iter += 1
              perm[i], perm[j] = perm[j], perm[i]
              score = _calc_score(A, B, perm)
              if better(score, best_score):
                  best_score = score
                  break
              # faster to swap back than to create a new list every time
              perm[i], perm[j] = perm[j], perm[i]
          else:  # no swaps made
              done = True
  
      res = {"col_ind": perm, "fun": best_score, "nit": n_iter}
      return OptimizeResult(res)
  