.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/graph_theory/assignment/plot_01_hungarian_method.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

    .. note::
        :class: sphx-glr-download-link-note

        :ref:`Go to the end <sphx_glr_download_api_gallery_graph_theory_assignment_plot_01_hungarian_method.py>`
        to download the full example code.

.. rst-class:: sphx-glr-example-title

.. _sphx_glr_api_gallery_graph_theory_assignment_plot_01_hungarian_method.py:


Kuhn's Hungarian method: the assignment problem
=====================================================

Assigns workers to jobs at minimum total cost. Brute force would check
n! assignments; the Hungarian method solves the problem in polynomial
time, and matches brute force on a small instance.

.. GENERATED FROM PYTHON SOURCE LINES 11-19

.. code-block:: Python

    import time
    from itertools import permutations

    import matplotlib.pyplot as plt
    import numpy as np

    from mathematicskit.graph_theory import solve_assignment








.. GENERATED FROM PYTHON SOURCE LINES 20-22

A small instance, checked by brute force
-----------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 22-37

.. code-block:: Python


    rng = np.random.default_rng(0)
    cost = rng.integers(1, 30, size=(6, 6))
    result = solve_assignment(cost)
    brute = min(sum(cost[i, p[i]] for i in range(6)) for p in permutations(range(6)))
    print(cost)
    print(f"optimal assignment: {dict(zip(result.rows.tolist(), result.cols.tolist()))}, cost {result.total_cost} (brute force {brute})")

    fig, ax = plt.subplots()
    ax.imshow(cost, cmap="Blues")
    ax.plot(result.cols, result.rows, "rx", ms=14, mew=3)
    ax.set_xlabel("job")
    ax.set_ylabel("worker")
    ax.set_title("Chosen entries (red): one per row and column")




.. image-sg:: /api/gallery/graph_theory/assignment/images/sphx_glr_plot_01_hungarian_method_001.png
   :alt: Chosen entries (red): one per row and column
   :srcset: /api/gallery/graph_theory/assignment/images/sphx_glr_plot_01_hungarian_method_001.png
   :class: sphx-glr-single-img


.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    [[25 19 15  8  9  2]
     [ 3  1  6 24 19 27]
     [15 18 29 22 19 16]
     [17 28  9 24 20  1]
     [12 25 17  1 23 22]
     [25  6  3 26  1 16]]
    optimal assignment: {0: 5, 1: 1, 2: 0, 3: 2, 4: 3, 5: 4}, cost 29.0 (brute force 29)

    Text(0.5, 1.0, 'Chosen entries (red): one per row and column')



.. GENERATED FROM PYTHON SOURCE LINES 38-40

Large instances are fast
-----------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 40-46

.. code-block:: Python


    for n in (100, 400, 1600):
        big = rng.random((n, n))
        start = time.perf_counter()
        total = solve_assignment(big).total_cost
        print(f"n = {n:4d}: optimal cost {total:.3f} in {time.perf_counter() - start:.3f} s ({n}! assignments)")




.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    n =  100: optimal cost 1.721 in 0.000 s (100! assignments)
    n =  400: optimal cost 1.604 in 0.003 s (400! assignments)
    n = 1600: optimal cost 1.651 in 0.074 s (1600! assignments)





.. rst-class:: sphx-glr-timing

   **Total running time of the script:** (0 minutes 0.107 seconds)


.. _sphx_glr_download_api_gallery_graph_theory_assignment_plot_01_hungarian_method.py:

.. only:: html

  .. container:: sphx-glr-footer sphx-glr-footer-example

    .. container:: sphx-glr-download sphx-glr-download-jupyter

      :download:`Download Jupyter notebook: plot_01_hungarian_method.ipynb <plot_01_hungarian_method.ipynb>`

    .. container:: sphx-glr-download sphx-glr-download-python

      :download:`Download Python source code: plot_01_hungarian_method.py <plot_01_hungarian_method.py>`

    .. container:: sphx-glr-download sphx-glr-download-zip

      :download:`Download zipped: plot_01_hungarian_method.zip <plot_01_hungarian_method.zip>`


.. only:: html

 .. rst-class:: sphx-glr-signature

    `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_