105 lines
3.9 KiB
Python
105 lines
3.9 KiB
Python
# 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
|
|
|
|
import numpy as np
|
|
from . import _lsap_module
|
|
|
|
|
|
def linear_sum_assignment(cost_matrix, maximize=False):
|
|
"""Solve the linear sum assignment problem.
|
|
|
|
The linear sum assignment problem 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.
|
|
|
|
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])``.
|
|
|
|
Notes
|
|
-----
|
|
.. 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, https://doi.org/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
|
|
"""
|
|
cost_matrix = np.asarray(cost_matrix)
|
|
if len(cost_matrix.shape) != 2:
|
|
raise ValueError("expected a matrix (2-D array), got a %r array"
|
|
% (cost_matrix.shape,))
|
|
|
|
if not (np.issubdtype(cost_matrix.dtype, np.number) or
|
|
cost_matrix.dtype == np.dtype(np.bool_)):
|
|
raise ValueError("expected a matrix containing numerical entries, got %s"
|
|
% (cost_matrix.dtype,))
|
|
|
|
if maximize:
|
|
cost_matrix = -cost_matrix
|
|
|
|
if np.any(np.isneginf(cost_matrix) | np.isnan(cost_matrix)):
|
|
raise ValueError("matrix contains invalid numeric entries")
|
|
|
|
cost_matrix = cost_matrix.astype(np.double)
|
|
a = np.arange(np.min(cost_matrix.shape))
|
|
|
|
# The algorithm expects more columns than rows in the cost matrix.
|
|
if cost_matrix.shape[1] < cost_matrix.shape[0]:
|
|
b = _lsap_module.calculate_assignment(cost_matrix.T)
|
|
indices = np.argsort(b)
|
|
return (b[indices], a[indices])
|
|
else:
|
|
b = _lsap_module.calculate_assignment(cost_matrix)
|
|
return (a, b)
|