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Module « scipy.optimize »

Fonction linear_sum_assignment - module scipy.optimize

Signature de la fonction linear_sum_assignment

Description

help(scipy.optimize.linear_sum_assignment)

Solve the linear sum assignment problem.

Parameters
----------
cost_matrix : array
    The cost matrix of the bipartite graph.

maximize : bool (default: False)
    Calculates a maximum weight matching if true.

Returns
-------
row_ind, col_ind : array
    An array of row indices and one of corresponding column indices giving
    the optimal assignment. The cost of the assignment can be computed
    as ``cost_matrix[row_ind, col_ind].sum()``. The row indices will be
    sorted; in the case of a square cost matrix they will be equal to
    ``numpy.arange(cost_matrix.shape[0])``.

See Also
--------
scipy.sparse.csgraph.min_weight_full_bipartite_matching : for sparse inputs

Notes
-----

The linear sum assignment problem [1]_ is also known as minimum weight
matching in bipartite graphs. A problem instance is described by a matrix
C, where each C[i,j] is the cost of matching vertex i of the first partite
set (a 'worker') and vertex j of the second set (a 'job'). The goal is to
find a complete assignment of workers to jobs of minimal cost.

Formally, let X be a boolean matrix where :math:`X[i,j] = 1` iff row i is
assigned to column j. Then the optimal assignment has cost

.. math::
    \min \sum_i \sum_j C_{i,j} X_{i,j}

where, in the case where the matrix X is square, each row is assigned to
exactly one column, and each column to exactly one row.

This function can also solve a generalization of the classic assignment
problem where the cost matrix is rectangular. If it has more rows than
columns, then not every row needs to be assigned to a column, and vice
versa.

This implementation is a modified Jonker-Volgenant algorithm with no
initialization, described in ref. [2]_.

.. versionadded:: 0.17.0

References
----------

.. [1] https://en.wikipedia.org/wiki/Assignment_problem

.. [2] DF Crouse. On implementing 2D rectangular assignment algorithms.
       *IEEE Transactions on Aerospace and Electronic Systems*,
       52(4):1679-1696, August 2016, :doi:`10.1109/TAES.2016.140952`

Examples
--------
>>> import numpy as np
>>> cost = np.array([[4, 1, 3], [2, 0, 5], [3, 2, 2]])
>>> from scipy.optimize import linear_sum_assignment
>>> row_ind, col_ind = linear_sum_assignment(cost)
>>> col_ind
array([1, 0, 2])
>>> cost[row_ind, col_ind].sum()
5

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