• Introduction
  • Assignment Problem
• Usage
• Calculating profit, rather than cost
• Non-square Cost Matrices
• Installation
• Source Code Repository
• Author
• Copyright
• License
• References

Introduction

The Munkres module provides an implementation of the Munkres algorithm (also called the Hungarian algorithm or the Kuhn-Munkres algorithm), useful for solving the Assignment Problem.

Assignment Problem

Let C be an nxn matrix representing the costs of each of n workers to perform any of n jobs. The assignment problem is to assign jobs to workers in a way that minimizes the total cost. Since each worker can perform only one job and each job can be assigned to only one worker, the assignments represent an independent set of the matrix C.

One way to generate the optimal set is to create all permutations of the indexes necessary to traverse the matrix so that no row and column are used more than once. For instance, given this matrix (expressed in Python):

matrix = [[5, 9, 1],
          [10, 3, 2],
          [8, 7, 4]]

def permute(a, results):
    if len(a) == 1:
        results.insert(len(results), a)

    else:
        for i in range(0, len(a)):
            element = a[i]
            a_copy = [a[j] for j in range(0, len(a)) if j != i]
            subresults = []
            permute(a_copy, subresults)
            for subresult in subresults:
                result = [element] + subresult
                results.insert(len(results), result)

results = []
permute(range(len(matrix)), results) # [0, 1, 2] for a 3x3 matrix

[[0, 1, 2],
 [0, 2, 1],
 [1, 0, 2],
 [1, 2, 0],
 [2, 0, 1],
 [2, 1, 0]]

n = len(matrix)
minval = sys.maxint
for row in range(n):
    cost = 0
    for col in range(n):
        cost += matrix[row][col]
    minval = min(cost, minval)

print minval

While this approach works fine for small matrices, it does not scale. It executes in O(n!) time: Calculating the permutations for an nxn matrix requires n! operations. For a 12x12 matrix, that's 479,001,600 traversals. Even if you could manage to perform each traversal in just one millisecond, it would still take more than 133 hours to perform the entire traversal. A 20x20 matrix would take 2,432,902,008,176,640,000 operations. At an optimistic one millisecond per operation, that's more than 77 million years.

The Munkres algorithm runs in O(n³) time, rather than O(n!). This package provides an implementation of that algorithm.

This version is based on http://www.public.iastate.edu/~ddoty/HungarianAlgorithm.html

This version was adapted for Python by Brian Clapper from the algorithm at the above web site. (The Algorithm::Munkres Perl version, in CPAN, was clearly adapted from the same web site.)

Usage

from munkres import Munkres

m = Munkres()

from munkres import Munkres, print_matrix

matrix = [[5, 9, 1],
          [10, 3, 2],
          [8, 7, 4]]
m = Munkres()
indexes = m.compute(matrix)
print_matrix(matrix, msg='Lowest cost through this matrix:')
total = 0
for row, column in indexes:
    value = matrix[row][column]
    total += value
    print '(%d, %d) -> %d' % (row, column, value)
print 'total cost: %d' % total

Running that program produces:

Lowest cost through this matrix:
[5, 9, 1]
[10, 3, 2]
[8, 7, 4]
(0, 0) -> 5
(1, 1) -> 3
(2, 2) -> 4
total cost=12

The Munkres object can be used multiple times.

Non-square Cost Matrices

• The module operates on a copy of the caller's matrix, so any padding will not be seen by the caller.
• The cost matrix must be rectangular or square. An irregular matrix will not work.

Calculating profit, rather than cost

The cost matrix is just that: A cost matrix. The Munkres algorithm finds the combination of elements (one from each row and column) that results in the smallest cost. It's also possible to use the algorithm to maximize profit. To do that, however, you have to convert your profit matrix to a cost matrix. The simplest way to do that is to subtract all elements from a large value. For example:

from munkres import Munkres, print_matrix

matrix = [[5, 9, 1],
          [10, 3, 2],
          [8, 7, 4]]
cost_matrix = []
for row in matrix:
    cost_row = []
    for col in row:
        cost_row += [sys.maxint - col]
    cost_matrix += [cost_row]

m = Munkres()
indexes = m.compute(cost_matrix)
print_matrix(matrix, msg='Highest profit through this matrix')
total = 0
for row, column in indexes:
    value = matrix[row][column]
    total += value
    print '(%d, %d) -> %d' % (row, column, value)

Print 'total profit=%d' % total

Highest profit through this matrix:
[5, 9, 1]
[10, 3, 2]
[8, 7, 4]
(0, 1) -> 9
(1, 0) -> 10
(2, 2) -> 4
total profit=23

cost_matrix = Munkres.make_cost_matrix(matrix,
                                       lambda cost: sys.maxint - cost)

from munkres import Munkres, print_matrix

matrix = [[5, 9, 1],
          [10, 3, 2],
          [8, 7, 4]]
cost_matrix = Munkres.make_cost_matrix(matrix,
                                       lambda cost: sys.maxint - cost)
m = Munkres()
indexes = m.compute(cost_matrix)
print_matrix(matrix, msg='Lowest cost through this matrix:')
total = 0
for row, column in indexes:
    value = matrix[row][column]
    total += value
    print '(%d, %d) -> %d' % (row, column, value)
print 'total profit=%d' % total

Installation

The usual:

1. Unpack the tarball or Zip file.
2. Change your directory to the resulting munkres-1.0.5.3 directory.
3. Run: python setup.py install

easy_install munkres

Source Code Repository

The source code for munkres is maintained on github. To clone the repository, run this command:

git clone git://github.com/bmc/munkres.git

Author

Brian M. Clapper, <bmc @ clapper . org>

Copyright

© 2008 Brian M. Clapper

License

BSD-style license. See accompanying LICENSE file.

References

1. http://www.public.iastate.edu/~ddoty/HungarianAlgorithm.html
2. Harold W. Kuhn. The Hungarian Method for the assignment problem. Naval Research Logistics Quarterly, 2:83-97, 1955.
3. Harold W. Kuhn. Variants of the Hungarian method for assignment problems. Naval Research Logistics Quarterly, 3: 253-258, 1956.
4. Munkres, J. Algorithms for the Assignment and Transportation Problems. Journal of the Society of Industrial and Applied Mathematics, 5(1):32-38, March, 1957.
5. http://en.wikipedia.org/wiki/Hungarian_algorithm