NAME

           Algorithm::Munkres - Perl extension for Munkres' solution to
           classical Assignment problem for square and rectangular matrices
           This module extends the solution of Assignment problem for square
           matrices to rectangular matrices by padding zeros. Thus a rectangular
           matrix is converted to square matrix by padding necessary zeros.

SYNOPSIS

       use Algorithm::Munkres;

           @mat = (
                [2, 4, 7, 9],
                [3, 9, 5, 1],
                [8, 2, 9, 7],
                );

       assign(\@mat,\@out_mat);

           Then the @out_mat array will have the output as: (0,3,1,2),
           where
           0th element indicates that 0th row is assigned 0th column i.e value=2
           1st element indicates that 1st row is assigned 3rd column i.e.value=1
           2nd element indicates that 2nd row is assigned 1st column.i.e.value=2
           3rd element indicates that 3rd row is assigned 2nd column.i.e.value=0

DESCRIPTION

           Assignment Problem: Given N jobs, N workers and the time taken by
           each worker to complete a job then how should the assignment of a
           Worker to a Job be done, so as to minimize the time taken.

               Thus if we have 3 jobs p,q,r and 3 workers x,y,z such that:
                   x  y  z
                p  2  4  7
                q  3  9  5
                r  8  2  9

               where the cell values of the above matrix give the time required
               for the worker(given by column name) to complete the job(given by
               the row name)

               then possible solutions are:
                                Total
                1. 2, 9, 9       20
                2. 2, 2, 5        9
                3. 3, 4, 9       16
                4. 3, 2, 7       12
                5. 8, 9, 7       24
                6. 8, 4, 5       17

           Thus (2) is the optimal solution for the above problem.
           This kind of brute-force approach of solving Assignment problem
           quickly becomes slow and bulky as N grows, because the number of
           possible solution are N! and thus the task is to evaluate each
           and then find the optimal solution.(If N=10, number of possible
           solutions: 3628800 !)
           Munkres' gives us a solution to this problem, which is implemented
           in this module.

           This module also solves Assignment problem for rectangular matrices
           (M x N) by converting them to square matrices by padding zeros. ex:
           If input matrix is:
                [2, 4, 7, 9],
                [3, 9, 5, 1],
                [8, 2, 9, 7]
           i.e 3 x 4 then we will convert it to 4 x 4 and the modified input
           matrix will be:
                [2, 4, 7, 9],
                [3, 9, 5, 1],
                [8, 2, 9, 7],
                [0, 0, 0, 0]

EXPORT

           "assign" function by default.

INPUT

           The input matrix should be in a two dimensional array(array of
           array) and the 'assign' subroutine expects a reference to this
           array and not the complete array.
           eg:assign(\@inp_mat, \@out_mat);
           The second argument to the assign subroutine is the reference
           to the output array.

OUTPUT

           The assign subroutine expects references to two arrays as its
           input paramenters. The second parameter is the reference to the
           output array. This array is populated by assign subroutine. This
           array is single dimensional Nx1 matrix.
           For above example the output array returned will be:
            (0,
            2,
            1)

           where
           0th element indicates that 0th row is assigned 0th column i.e value=2
           1st element indicates that 1st row is assigned 2nd column i.e.value=5
           2nd element indicates that 2nd row is assigned 1st column.i.e.value=2

SEE ALSO

           1. http://216.249.163.93/bob.pilgrim/445/munkres.html

           2. Munkres, J. Algorithms for the assignment and transportation
              Problems. J. Siam 5 (Mar. 1957), 32-38

           3. FranA~Xois Bourgeois and Jean-Claude Lassalle. 1971.
              An extension of the Munkres algorithm for the assignment
              problem to rectangular matrices.
              Communication ACM, 14(12):802-804

AUTHOR

           Anagha Kulkarni, University of Minnesota Duluth
           kulka020 <at> d.umn.edu

           Ted Pedersen, University of Minnesota Duluth
           tpederse <at> d.umn.edu

COPYRIGHT AND LICENSE

       Copyright (C) 2007-2008, Ted Pedersen and Anagha Kulkarni

       This program is free software; you can redistribute it and/or modify it under the terms of
       the GNU General Public License as published by the Free Software Foundation; either
       version 2 of the License, or (at your option) any later version.  This program is
       distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the
       implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
       General Public License for more details.

       You should have received a copy of the GNU General Public License along with this program;
       if not, write to the Free Software Foundation, Inc., 59 Temple Place - Suite 330, Boston,
       MA  02111-1307, USA.