Current File : //lib/python3/dist-packages/scipy/optimize/_lsap.py
# Wrapper for the shortest augmenting path algorithm for solving the
# rectangular linear sum assignment problem. The original code was an
# implementation of the Hungarian algorithm (Kuhn-Munkres) taken from
# scikit-learn, based on original code by Brian Clapper and adapted to NumPy
# by Gael Varoquaux. Further improvements by Ben Root, Vlad Niculae, Lars
# Buitinck, and Peter Larsen.
#
# Copyright (c) 2008 Brian M. Clapper <bmc@clapper.org>, Gael Varoquaux
# Author: Brian M. Clapper, Gael Varoquaux
# License: 3-clause BSD
from . import _lsap_module
def linear_sum_assignment(cost_matrix, maximize=False):
"""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
--------
>>> 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
"""
return _lsap_module.calculate_assignment(cost_matrix, maximize)
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