# pygmtools.linear_solvers.hungarian

pygmtools.linear_solvers.hungarian(s, n1=None, n2=None, unmatch1=None, unmatch2=None, nproc: int = 1, backend=None)[source]

Solve optimal LAP permutation by hungarian algorithm. The time cost is $$O(n^3)$$.

Parameters
• s$$(b\times n_1 \times n_2)$$ input 3d tensor. $$b$$: batch size. Non-batched input is also supported if s is of size $$(n_1 \times n_2)$$

• n1$$(b)$$ (optional) number of objects in dim1

• n2$$(b)$$ (optional) number of objects in dim2

• unmatch1 – (optional, new in 0.3.0) $$(b\times n_1)$$ the scores indicating the objects in dim1 is unmatched

• unmatch2 – (optional, new in 0.3.0) $$(b\times n_2)$$ the scores indicating the objects in dim2 is unmatched

• nproc – (default: 1, i.e. no parallel) number of parallel processes

• backend – (default: pygmtools.BACKEND variable) the backend for computation.

Returns

$$(b\times n_1 \times n_2)$$ optimal permutation matrix

Note

The parallelization is based on multi-processing workers that run on multiple CPU cores.

Note

For all backends, scipy.optimize.linear_sum_assignment is called to solve the LAP, therefore the computation is based on numpy and scipy. The backend argument of this function only affects the input-output data type.

Note

We support batched instances with different number of nodes, therefore n1 and n2 are required to specify the exact number of objects of each dimension in the batch. If not specified, we assume the batched matrices are not padded and all elements in n1 are equal, all in n2 are equal.

Warning

This function can work with or without maximal inlier matching:

• With maximal inlier matching (the default mode). If unmatch1=None and unmatch2=None, the solver aims to match as many nodes as possible. The corresponding linear assignment problem is

$\begin{split}&\max_{\mathbf{X}} \ \texttt{tr}(\mathbf{X}^\top \mathbf{S})\\ s.t. \quad &\mathbf{X} \in \{0, 1\}^{n_1\times n_2}, \ \mathbf{X}\mathbf{1} = \mathbf{1}, \ \mathbf{X}^\top\mathbf{1} \leq \mathbf{1}\end{split}$

where the constraint $$\mathbf{X}\mathbf{1} = \mathbf{1}$$ urges the solver to match as many inlier nodes as possible.

• Without maximal inlier matching (new in 0.3.0). If unmatch1 and unmatch2 are not None, the solver is allowed to match nodes to void nodes, and the corresponding scores for matching to void nodes are specified by unmatch1 and unmatch2. The following (modified) linear assignment problem is considered:

$\begin{split}&\max_{\mathbf{X}} \ \texttt{tr}(\mathbf{X}^\top \mathbf{S}^\prime)\\ s.t. \quad &\mathbf{X} \in \{0, 1\}^{n_1+1\times n_2+1}, \ \mathbf{X}_{[0:n_1, :]}\mathbf{1} = \mathbf{1}, \ \mathbf{X}_{[:, 0:n_2]}^\top\mathbf{1} \leq \mathbf{1}\end{split}$

where the last column and last row of $$\mathbf{S}^\prime$$ are unmatch1 and unmatch2, respectively.

For example, if you want to solve the following problem (note that both consrtraints are $$\leq$$)

$\begin{split}&\max_{\mathbf{X}} \ \texttt{tr}(\mathbf{X}^\top \mathbf{S})\\ s.t. \quad &\mathbf{X} \in \{0, 1\}^{n_1\times n_2}, \ \mathbf{X}\mathbf{1} \leq \mathbf{1}, \ \mathbf{X}^\top\mathbf{1} \leq \mathbf{1}\end{split}$

you can simply set unmatch1 and unmatch2 as zero vectors.

Numpy Example
>>> import numpy as np
>>> import pygmtools as pygm
>>> pygm.set_backend('numpy')
>>> np.random.seed(0)

# 2-dimensional (non-batched) input
>>> s_2d = np.random.rand(5, 5)
>>> s_2d
array([[0.5488135 , 0.71518937, 0.60276338, 0.54488318, 0.4236548 ],
[0.64589411, 0.43758721, 0.891773  , 0.96366276, 0.38344152],
[0.79172504, 0.52889492, 0.56804456, 0.92559664, 0.07103606],
[0.0871293 , 0.0202184 , 0.83261985, 0.77815675, 0.87001215],
[0.97861834, 0.79915856, 0.46147936, 0.78052918, 0.11827443]])

>>> x = pygm.hungarian(s_2d)
>>> x
array([[0., 1., 0., 0., 0.],
[0., 0., 1., 0., 0.],
[0., 0., 0., 1., 0.],
[0., 0., 0., 0., 1.],
[1., 0., 0., 0., 0.]])

# 3-dimensional (batched) input
>>> s_3d = np.random.rand(3, 5, 5)
>>> n1 = n2 = np.array([3, 4, 5])
>>> x = pygm.hungarian(s_3d, n1, n2)
>>> x
array([[[0., 0., 1., 0., 0.],
[0., 1., 0., 0., 0.],
[1., 0., 0., 0., 0.],
[0., 0., 0., 0., 0.],
[0., 0., 0., 0., 0.]],

[[1., 0., 0., 0., 0.],
[0., 1., 0., 0., 0.],
[0., 0., 1., 0., 0.],
[0., 0., 0., 1., 0.],
[0., 0., 0., 0., 0.]],

[[0., 0., 1., 0., 0.],
[1., 0., 0., 0., 0.],
[0., 0., 0., 0., 1.],
[0., 1., 0., 0., 0.],
[0., 0., 0., 1., 0.]]])

# allow matching to void nodes by setting unmatch1 and unmatch2
>>> s_2d = np.random.randn(5, 5)
>>> s_2d
array([[-1.16514984,  0.90082649,  0.46566244, -1.53624369,  1.48825219],
[ 1.89588918,  1.17877957, -0.17992484, -1.07075262,  1.05445173],
[-0.40317695,  1.22244507,  0.20827498,  0.97663904,  0.3563664 ],
[ 0.70657317,  0.01050002,  1.78587049,  0.12691209,  0.40198936],
[ 1.8831507 , -1.34775906, -1.270485  ,  0.96939671, -1.17312341]])
>>> unmatch1 = np.random.randn(5)
>>> unmatch1
array([ 1.94362119, -0.41361898, -0.74745481,  1.92294203,  1.48051479])
>>> unmatch2 = np.random.randn(5)
>>> unmatch2
array([ 1.86755896,  0.90604466, -0.86122569,  1.91006495, -0.26800337])
>>> x = pygm.hungarian(s_2d, unmatch1=unmatch1, unmatch2=unmatch2)
>>> x
array([[0., 0., 0., 0., 0.],
[0., 0., 0., 0., 1.],
[0., 0., 1., 0., 0.],
[0., 0., 0., 0., 0.],
[0., 0., 0., 0., 0.]])
Pytorch Example
>>> import torch
>>> import pygmtools as pygm
>>> pygm.set_backend('pytorch')
>>> np.random.seed(0)

# 2-dimensional (non-batched) input
>>> s_2d = torch.from_numpy(np.random.rand(5, 5))
>>> s_2d
tensor([[0.5488, 0.7152, 0.6028, 0.5449, 0.4237],
[0.6459, 0.4376, 0.8918, 0.9637, 0.3834],
[0.7917, 0.5289, 0.5680, 0.9256, 0.0710],
[0.0871, 0.0202, 0.8326, 0.7782, 0.8700],
[0.9786, 0.7992, 0.4615, 0.7805, 0.1183]], dtype=torch.float64)
>>> x = pygm.hungarian(s_2d)
>>> x
tensor([[0., 1., 0., 0., 0.],
[0., 0., 1., 0., 0.],
[0., 0., 0., 1., 0.],
[0., 0., 0., 0., 1.],
[1., 0., 0., 0., 0.]], dtype=torch.float64)

# 3-dimensional (batched) input
>>> s_3d = torch.from_numpy(np.random.rand(3, 5, 5))
>>> n1 = n2 = torch.tensor([3, 4, 5])
>>> x = pygm.hungarian(s_3d, n1, n2)
>>> x
tensor([[[0., 0., 1., 0., 0.],
[0., 1., 0., 0., 0.],
[1., 0., 0., 0., 0.],
[0., 0., 0., 0., 0.],
[0., 0., 0., 0., 0.]],

[[1., 0., 0., 0., 0.],
[0., 1., 0., 0., 0.],
[0., 0., 1., 0., 0.],
[0., 0., 0., 1., 0.],
[0., 0., 0., 0., 0.]],

[[0., 0., 1., 0., 0.],
[1., 0., 0., 0., 0.],
[0., 0., 0., 0., 1.],
[0., 1., 0., 0., 0.],
[0., 0., 0., 1., 0.]]], dtype=torch.float64)

# allow matching to void nodes by setting unmatch1 and unmatch2
>>> s_2d = torch.from_numpy(np.random.randn(5, 5))
>>> s_2d
tensor([[-1.1651,  0.9008,  0.4657, -1.5362,  1.4883],
[ 1.8959,  1.1788, -0.1799, -1.0708,  1.0545],
[-0.4032,  1.2224,  0.2083,  0.9766,  0.3564],
[ 0.7066,  0.0105,  1.7859,  0.1269,  0.4020],
[ 1.8832, -1.3478, -1.2705,  0.9694, -1.1731]], dtype=torch.float64)
>>> unmatch1 = torch.from_numpy(np.random.randn(5))
>>> unmatch1
tensor([ 1.9436, -0.4136, -0.7475,  1.9229,  1.4805], dtype=torch.float64)
>>> unmatch2 = torch.from_numpy(np.random.randn(5))
>>> unmatch2
tensor([ 1.8676,  0.9060, -0.8612,  1.9101, -0.2680], dtype=torch.float64)
>>> x = pygm.hungarian(s_2d, unmatch1=unmatch1, unmatch2=unmatch2)
>>> x
tensor([[0., 0., 0., 0., 0.],
[0., 0., 0., 0., 1.],
[0., 0., 1., 0., 0.],
[0., 0., 0., 0., 0.],
[0., 0., 0., 0., 0.]], dtype=torch.float64)
>>> import pygmtools as pygm
>>> np.random.seed(0)

# 2-dimensional (non-batched) input
>>> s_2d
[[0.54881350, 0.71518937, 0.60276338, 0.54488318, 0.42365480],
[0.64589411, 0.43758721, 0.89177300, 0.96366276, 0.38344152],
[0.79172504, 0.52889492, 0.56804456, 0.92559664, 0.07103606],
[0.08712930, 0.02021840, 0.83261985, 0.77815675, 0.87001215],
[0.97861834, 0.79915856, 0.46147936, 0.78052918, 0.11827443]])
>>> x = pygm.hungarian(s_2d)
>>> x
[[0., 1., 0., 0., 0.],
[0., 0., 1., 0., 0.],
[0., 0., 0., 1., 0.],
[0., 0., 0., 0., 1.],
[1., 0., 0., 0., 0.]])

# 3-dimensional (batched) input
>>> s_3d = paddle.to_tensor(np.random.rand(3, 5, 5))
>>> n1 = n2 = paddle.to_tensor([3, 4, 5])
>>> x = pygm.hungarian(s_3d, n1, n2)
>>> x
Tensor(shape=[3, 5, 5], dtype=float64, place=Place(cpu), stop_gradient=True,
[[[0., 0., 1., 0., 0.],
[0., 1., 0., 0., 0.],
[1., 0., 0., 0., 0.],
[0., 0., 0., 0., 0.],
[0., 0., 0., 0., 0.]],

[[1., 0., 0., 0., 0.],
[0., 1., 0., 0., 0.],
[0., 0., 1., 0., 0.],
[0., 0., 0., 1., 0.],
[0., 0., 0., 0., 0.]],

[[0., 0., 1., 0., 0.],
[1., 0., 0., 0., 0.],
[0., 0., 0., 0., 1.],
[0., 1., 0., 0., 0.],
[0., 0., 0., 1., 0.]]])

# allow matching to void nodes by setting unmatch1 and unmatch2
>>> s_2d
[[-1.16514984,  0.90082649,  0.46566244, -1.53624369,  1.48825219],
[ 1.89588918,  1.17877957, -0.17992484, -1.07075262,  1.05445173],
[-0.40317695,  1.22244507,  0.20827498,  0.97663904,  0.35636640],
[ 0.70657317,  0.01050002,  1.78587049,  0.12691209,  0.40198936],
[ 1.88315070, -1.34775906, -1.27048500,  0.96939671, -1.17312341]])
>>> unmatch1
[ 1.94362119, -0.41361898, -0.74745481,  1.92294203,  1.48051479])
>>> unmatch2
[ 1.86755896,  0.90604466, -0.86122569,  1.91006495, -0.26800337])
>>> x = pygm.hungarian(s_2d, unmatch1=unmatch1, unmatch2=unmatch2)
>>> x
[[0., 0., 0., 0., 0.],
[0., 0., 0., 0., 1.],
[0., 0., 1., 0., 0.],
[0., 0., 0., 0., 0.],
[0., 0., 0., 0., 0.]])
Jittor Example
>>> import jittor as jt
>>> import pygmtools as pygm
>>> pygm.set_backend('jittor')
>>> np.random.seed(0)

# 2-dimensional (non-batched) input
>>> s_2d = pygm.utils.from_numpy(np.random.rand(5, 5))
>>> s_2d
jt.Var([[0.5488135  0.71518934 0.60276335 0.5448832  0.4236548 ]
[0.6458941  0.4375872  0.891773   0.96366274 0.3834415 ]
[0.79172504 0.5288949  0.56804454 0.92559665 0.07103606]
[0.0871293  0.0202184  0.83261985 0.77815676 0.87001216]
[0.9786183  0.7991586  0.46147937 0.7805292  0.11827443]], dtype=float32)
>>> x = pygm.hungarian(s_2d)
>>> x
jt.Var([[0. 1. 0. 0. 0.]
[0. 0. 1. 0. 0.]
[0. 0. 0. 1. 0.]
[0. 0. 0. 0. 1.]
[1. 0. 0. 0. 0.]], dtype=float32)

# 3-dimensional (batched) input
>>> s_3d = pygm.utils.from_numpy(np.random.rand(3, 5, 5))
>>> n1 = n2 = jt.Var([3, 4, 5])
>>> x = pygm.hungarian(s_3d, n1, n2)
>>> x
jt.Var([[[0. 0. 1. 0. 0.]
[0. 1. 0. 0. 0.]
[1. 0. 0. 0. 0.]
[0. 0. 0. 0. 0.]
[0. 0. 0. 0. 0.]]

[[1. 0. 0. 0. 0.]
[0. 1. 0. 0. 0.]
[0. 0. 1. 0. 0.]
[0. 0. 0. 1. 0.]
[0. 0. 0. 0. 0.]]

[[0. 0. 1. 0. 0.]
[1. 0. 0. 0. 0.]
[0. 0. 0. 0. 1.]
[0. 1. 0. 0. 0.]
[0. 0. 0. 1. 0.]]], dtype=float32)

# allow matching to void nodes by setting unmatch1 and unmatch2
>>> s_2d = pygm.utils.from_numpy(np.random.randn(5, 5))
>>> s_2d
jt.Var([[-1.1651498   0.9008265   0.46566245 -1.5362437   1.4882522 ]
[ 1.8958892   1.1787796  -0.17992483 -1.0707526   1.0544517 ]
[-0.40317693  1.222445    0.20827498  0.97663903  0.3563664 ]
[ 0.7065732   0.01050002  1.7858706   0.12691209  0.40198937]
[ 1.8831507  -1.347759   -1.270485    0.9693967  -1.1731234 ]], dtype=float32)
>>> unmatch1 = pygm.utils.from_numpy(np.random.randn(5))
>>> unmatch1
jt.Var([ 1.9436212  -0.41361898 -0.7474548   1.922942    1.4805148 ], dtype=float32)
>>> unmatch2 = pygm.utils.from_numpy(np.random.randn(5))
>>> unmatch2
jt.Var([ 1.867559    0.90604466 -0.86122566  1.9100649  -0.26800337], dtype=float32)
>>> x = pygm.hungarian(s_2d, unmatch1=unmatch1, unmatch2=unmatch2)
>>> x
jt.Var([[0. 0. 0. 0. 0.]
[0. 0. 0. 0. 1.]
[0. 0. 1. 0. 0.]
[0. 0. 0. 0. 0.]
[0. 0. 0. 0. 0.]], dtype=float32)
MindSpore Example
>>> import mindspore
>>> import pygmtools as pygm
>>> pygm.set_backend('mindspore')
>>> np.random.seed(0)

# 2-dimensional (non-batched) input
>>> s_2d = mindspore.Tensor(np.random.rand(5, 5))
>>> s_2d
Tensor(shape=[5, 5], dtype=Float64, value=
[[5.48813504e-001, 7.15189366e-001, 6.02763376e-001, 5.44883183e-001, 4.23654799e-001],
[6.45894113e-001, 4.37587211e-001, 8.91773001e-001, 9.63662761e-001, 3.83441519e-001],
[7.91725038e-001, 5.28894920e-001, 5.68044561e-001, 9.25596638e-001, 7.10360582e-002],
[8.71292997e-002, 2.02183974e-002, 8.32619846e-001, 7.78156751e-001, 8.70012148e-001],
[9.78618342e-001, 7.99158564e-001, 4.61479362e-001, 7.80529176e-001, 1.18274426e-001]])
>>> x = pygm.hungarian(s_2d)
>>> x
Tensor(shape=[5, 5], dtype=Float64, value=
[[0.00000000e+000, 1.00000000e+000, 0.00000000e+000, 0.00000000e+000, 0.00000000e+000],
[0.00000000e+000, 0.00000000e+000, 1.00000000e+000, 0.00000000e+000, 0.00000000e+000],
[0.00000000e+000, 0.00000000e+000, 0.00000000e+000, 1.00000000e+000, 0.00000000e+000],
[0.00000000e+000, 0.00000000e+000, 0.00000000e+000, 0.00000000e+000, 1.00000000e+000],
[1.00000000e+000, 0.00000000e+000, 0.00000000e+000, 0.00000000e+000, 0.00000000e+000]])

# 3-dimensional (batched) input
>>> s_3d = mindspore.Tensor(np.random.rand(3, 5, 5))
>>> n1 = n2 = mindspore.Tensor([3, 4, 5])
>>> x = pygm.hungarian(s_3d, n1, n2)
>>> x
Tensor(shape=[3, 5, 5], dtype=Float64, value=
[[[0.00000000e+000, 0.00000000e+000, 1.00000000e+000, 0.00000000e+000, 0.00000000e+000],
[0.00000000e+000, 1.00000000e+000, 0.00000000e+000, 0.00000000e+000, 0.00000000e+000],
[1.00000000e+000, 0.00000000e+000, 0.00000000e+000, 0.00000000e+000, 0.00000000e+000],
[0.00000000e+000, 0.00000000e+000, 0.00000000e+000, 0.00000000e+000, 0.00000000e+000],
[0.00000000e+000, 0.00000000e+000, 0.00000000e+000, 0.00000000e+000, 0.00000000e+000]],
[[1.00000000e+000, 0.00000000e+000, 0.00000000e+000, 0.00000000e+000, 0.00000000e+000],
[0.00000000e+000, 1.00000000e+000, 0.00000000e+000, 0.00000000e+000, 0.00000000e+000],
[0.00000000e+000, 0.00000000e+000, 1.00000000e+000, 0.00000000e+000, 0.00000000e+000],
[0.00000000e+000, 0.00000000e+000, 0.00000000e+000, 1.00000000e+000, 0.00000000e+000],
[0.00000000e+000, 0.00000000e+000, 0.00000000e+000, 0.00000000e+000, 0.00000000e+000]],
[[0.00000000e+000, 0.00000000e+000, 1.00000000e+000, 0.00000000e+000, 0.00000000e+000],
[1.00000000e+000, 0.00000000e+000, 0.00000000e+000, 0.00000000e+000, 0.00000000e+000],
[0.00000000e+000, 0.00000000e+000, 0.00000000e+000, 0.00000000e+000, 1.00000000e+000],
[0.00000000e+000, 1.00000000e+000, 0.00000000e+000, 0.00000000e+000, 0.00000000e+000],
[0.00000000e+000, 0.00000000e+000, 0.00000000e+000, 1.00000000e+000, 0.00000000e+000]]])

# allow matching to void nodes by setting unmatch1 and unmatch2
>>> s_2d = mindspore.Tensor(np.random.randn(5, 5))
>>> s_2d
Tensor(shape=[5, 5], dtype=Float64, value=
[[-1.16514984e+000, 9.00826487e-001, 4.65662440e-001, -1.53624369e+000, 1.48825219e+000],
[1.89588918e+000, 1.17877957e+000, -1.79924836e-001, -1.07075262e+000, 1.05445173e+000],
[-4.03176947e-001, 1.22244507e+000, 2.08274978e-001, 9.76639036e-001, 3.56366397e-001],
[7.06573168e-001, 1.05000207e-002, 1.78587049e+000, 1.26912093e-001, 4.01989363e-001],
[1.88315070e+000, -1.34775906e+000, -1.27048500e+000, 9.69396708e-001, -1.17312341e+000]])
>>> unmatch1 = mindspore.Tensor(np.random.randn(5))
>>> unmatch1
Tensor(shape=[5], dtype=Float64, value= [1.94362119e+000, -4.13618981e-001, -7.47454811e-001, 1.92294203e+000, 1.48051479e+000])
>>> unmatch2 = mindspore.Tensor(np.random.randn(5))
>>> unmatch2
Tensor(shape=[5], dtype=Float64, value= [1.86755896e+000, 9.06044658e-001, -8.61225685e-001, 1.91006495e+000, -2.68003371e-001])
>>> x = pygm.hungarian(s_2d, unmatch1=unmatch1, unmatch2=unmatch2)
>>> x
Tensor(shape=[5, 5], dtype=Float64, value=
[[0.00000000e+000, 0.00000000e+000, 0.00000000e+000, 0.00000000e+000, 0.00000000e+000],
[0.00000000e+000, 0.00000000e+000, 0.00000000e+000, 0.00000000e+000, 1.00000000e+000],
[0.00000000e+000, 0.00000000e+000, 1.00000000e+000, 0.00000000e+000, 0.00000000e+000],
[0.00000000e+000, 0.00000000e+000, 0.00000000e+000, 0.00000000e+000, 0.00000000e+000],
[0.00000000e+000, 0.00000000e+000, 0.00000000e+000, 0.00000000e+000, 0.00000000e+000]])
Tensorflow Example
>>> import tensorflow as tf
>>> import pygmtools as pygm
>>> pygm.set_backend('tensorflow')
>>> np.random.seed(0)

# 2-dimensional (non-batched) input
>>> s_2d = tf.constant(np.random.rand(5, 5))
>>> s_2d
<tf.Tensor: shape=(5, 5), dtype=float64, numpy=
array([[0.5488135 , 0.71518937, 0.60276338, 0.54488318, 0.4236548 ],
[0.64589411, 0.43758721, 0.891773  , 0.96366276, 0.38344152],
[0.79172504, 0.52889492, 0.56804456, 0.92559664, 0.07103606],
[0.0871293 , 0.0202184 , 0.83261985, 0.77815675, 0.87001215],
[0.97861834, 0.79915856, 0.46147936, 0.78052918, 0.11827443]])>
>>> x = pygm.hungarian(s_2d)
>>> x
<tf.Tensor: shape=(5, 5), dtype=float64, numpy=
array([[0., 1., 0., 0., 0.],
[0., 0., 1., 0., 0.],
[0., 0., 0., 1., 0.],
[0., 0., 0., 0., 1.],
[1., 0., 0., 0., 0.]])>

# 3-dimensional (batched) input
>>> s_3d = tf.constant(np.random.rand(3, 5, 5))
>>> n1 = n2 = tf.constant([3, 4, 5])
>>> x = pygm.hungarian(s_3d, n1, n2)
>>> x
<tf.Tensor: shape=(3, 5, 5), dtype=float64, numpy=
array([[[0., 0., 1., 0., 0.],
[0., 1., 0., 0., 0.],
[1., 0., 0., 0., 0.],
[0., 0., 0., 0., 0.],
[0., 0., 0., 0., 0.]],

[[1., 0., 0., 0., 0.],
[0., 1., 0., 0., 0.],
[0., 0., 1., 0., 0.],
[0., 0., 0., 1., 0.],
[0., 0., 0., 0., 0.]],

[[0., 0., 1., 0., 0.],
[1., 0., 0., 0., 0.],
[0., 0., 0., 0., 1.],
[0., 1., 0., 0., 0.],
[0., 0., 0., 1., 0.]]])>

# allow matching to void nodes by setting unmatch1 and unmatch2
>>> s_2d = tf.constant(np.random.randn(5, 5))
>>> s_2d
<tf.Tensor: shape=(5, 5), dtype=float64, numpy=
array([[-1.16514984,  0.90082649,  0.46566244, -1.53624369,  1.48825219],
[ 1.89588918,  1.17877957, -0.17992484, -1.07075262,  1.05445173],
[-0.40317695,  1.22244507,  0.20827498,  0.97663904,  0.3563664 ],
[ 0.70657317,  0.01050002,  1.78587049,  0.12691209,  0.40198936],
[ 1.8831507 , -1.34775906, -1.270485  ,  0.96939671, -1.17312341]])>
>>> unmatch1 = tf.constant(np.random.randn(5))
>>> unmatch1
<tf.Tensor: shape=(5,), dtype=float64, numpy=array([ 1.94362119, -0.41361898, -0.74745481,  1.92294203,  1.48051479])>
>>> unmatch2 = tf.constant(np.random.randn(5))
>>> unmatch2
<tf.Tensor: shape=(5,), dtype=float64, numpy=array([ 1.86755896,  0.90604466, -0.86122569,  1.91006495, -0.26800337])>
>>> x = pygm.hungarian(s_2d, unmatch1=unmatch1, unmatch2=unmatch2)
>>> x
<tf.Tensor: shape=(5, 5), dtype=float64, numpy=
array([[0., 0., 0., 0., 0.],
[0., 0., 0., 0., 1.],
[0., 0., 1., 0., 0.],
[0., 0., 0., 0., 0.],
[0., 0., 0., 0., 0.]])>

Note

If you find this graph matching solver useful for your research, please cite:

@article{hungarian,
title={Algorithms for the assignment and transportation problems},
author={Munkres, James},
journal={Journal of the society for industrial and applied mathematics},
volume={5},
number={1},
pages={32--38},
year={1957},
publisher={SIAM}
}