import numpy as np import operator import warnings import numbers 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' ` (default) and :ref:`'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 `_ 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') `. 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' ` uses the Fast Approximate QAP algorithm [1]_; it typically offers the best combination of speed and accuracy. Method :ref:`'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' `, 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' ` 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 """ 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}.") _spec007_transition(options.get("rng", None)) res = methods[method](A, B, **options) return res def _spec007_transition(rng): if isinstance(rng, np.random.RandomState): warnings.warn( "Use of `RandomState` with `quadratic_assignment` is deprecated" " and will result in an exception in SciPy 1.17", DeprecationWarning, stacklevel=2 ) if ((rng is None or rng is np.random) and np.random.mtrand._rand._bit_generator._seed_seq is None): warnings.warn( "The NumPy global RNG was seeded by calling `np.random.seed`." " From SciPy 1.17, this function will no longer use the global RNG.", FutureWarning, stacklevel=2 ) if isinstance(rng, numbers.Integral | np.integer): warnings.warn( "The behavior when the rng option is an integer is changing: the value" " will be normalized using np.random.default_rng beginning in SciPy 1.17," " and the resulting Generator will be used to generate random numbers.", FutureWarning, stacklevel=2 ) 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' ` 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`}, optional Pseudorandom number generator state. See `quadratic_assignment` for details. 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 scipy.optimize import quadratic_assignment >>> import numpy as np >>> rng = np.random.default_rng() >>> n = 15 >>> A = rng.random((n, n)) >>> B = rng.random((n, n)) >>> options = {"rng": rng} >>> res = quadratic_assignment(A, B, options=options) # FAQ is default method >>> print(res.fun) 47.797048706380636 # may vary >>> options = {"rng": rng, "P0": "randomized"} # use randomized initialization >>> res = quadratic_assignment(A, B, options=options) >>> print(res.fun) 47.37287069769966 # 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) 46.55974835248574 # may vary The '2-opt' method can be used to attempt to refine the results. >>> options = {"partial_guess": np.array([np.arange(n), res.col_ind]).T, "rng": rng} >>> res = quadratic_assignment(A, B, method="2opt", options=options) >>> print(res.fun) 46.55974835248574 # 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' ` is also available. Options ------- maximize : bool (default: False) Maximizes the objective function if ``True``. rng : {None, int, `numpy.random.Generator`}, optional Pseudorandom number generator state. See `quadratic_assignment` for details. 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`. .. note:: `partial_match` must be sorted by the first column. 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`. .. note:: `partial_guess` must be sorted by the first column. 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)