NAME

	Algorithm::Bertsekas - auction algorithm for the assignment problem.
	
	This is a perl implementation for the auction algorithm for the asymmetric (N<=M) assignment problem.

DESCRIPTION
 
 The assignment problem in the general form can be stated as follows:

 "Given N jobs (or persons), M tasks (or objects) and the effectiveness of each job for each task, 
 the problem is to assign each job to one and only one task in such a way that the measure of 
 effectiveness is optimised (Maximised or Minimised)."
 
 "Each assignment problem has associated with a table or matrix. Generally, the rows contain the 
 jobs (or persons) we wish to assign, and the columns comprise the tasks (or objects) we want them 
 assigned to. The numbers in the table are the costs associated with each particular assignment."
 
 In Auction Algorithm (AA) the N persons iteratively submit the bids to M objects.
 The AA take cost Matrix N×M = [aij] as an input and produce assignment as an output.
 In the AA persons iteratively submit the bids to the objects which are then reassigned 
 to the bidders which offer them the best bid.
 
 Another application is to find the (nearest/more distant) neighbors. 
 The distance between neighbors can be represented by a matrix or a weight function, for example:
 1: f(i,j) = abs ($array1[i] - $array2[j])
 2: f(i,j) = ($array1[i] - $array2[j]) ** 2
 

SYNOPSIS

 use Algorithm::Bertsekas qw(auction);
 
 Example 1: Find the nearest neighbor, Minimize the total benefit.

 my @array1 = ( 893, 401, 902, 576, 767, 917, 76, 464, 124, 207, 125, 530 );
 my @array2 = ( 161, 559, 247, 478, 456 );

 my @input_matrix;
 for my $i ( 0 .. $#array1 ){
   my @weight_function;		   
   for my $j ( 0 .. $#array2 ){
      my $weight = abs ($array1[$i] - $array2[$j]);
      #  $weight =     ($array1[$i] - $array2[$j]) ** 2;  # another option
      push @weight_function, $weight;
   }
   push @input_matrix, \@weight_function;
 }
  
       161 559 247 478 456

 893 [ 732 334 646 415 437 ]
 401 [ 240 158 154  77  55 ]
 902 [ 741 343 655 424 446 ]
 576 [ 415  17 329  98 120 ]
 767 [ 606 208 520 289 311 ]
 917 [ 756 358 670 439 461 ]
  76 [  85 483 171 402 380 ]
 464 [ 303  95 217  14   8 ]
 124 [  37 435 123 354 332 ]
 207 [  46 352  40 271 249 ]
 125 [  36 434 122 353 331 ]
 530 [ 369  29 283  52  74 ]
 
 my ( $optimal, $assignment_ref, $output_index_ref ) = auction( matrix_ref => \@input_matrix, maximize_total_benefit => 0, verbose => 5 );
  
 Objective: to Minimize the total benefit
 Number of left nodes: 12
 Number of right nodes: 5
 Number of edges: 60

 Solution:
 Optimal assignment: sum of values = 153
 Feasible assignment condition: stepsize = 0.1667 < 1/5 = 0.2
 Number of iterations: 50

 row index    = [  0   1   2   3   4   5   6   7   8   9  10  11 ]
 column index = [  9   8  10   1   5  11   7   4   6   2   0   3 ]
 matrix value = [             17               8      40  36  52 ]

 modified matrix 5 x 9:
 [ 516   341   150   671   453   719   710   720** 387  ]
 [ 598   739** 548   273   661   321   404   322   727  ]
 [ 602   427   236   585   539   633   716** 634   473  ]
 [ 679   658   467   354   742   402   485   403   704**]
 [ 701   636   445   376   748** 424   507   425   682  ]

 original matrix 12 x 5 with solution:
 [ 732   334   646   415   437  ]
 [ 240   158   154    77    55  ]
 [ 741   343   655   424   446  ]
 [ 415    17** 329    98   120  ]
 [ 606   208   520   289   311  ]
 [ 756   358   670   439   461  ]
 [  85   483   171   402   380  ]
 [ 303    95   217    14     8**]
 [  37   435   123   354   332  ]
 [  46   352    40** 271   249  ]
 [  36** 434   122   353   331  ]
 [ 369    29   283    52**  74  ]

 Pairs (in ascending order of matrix values):
   indices (  7,  4 ), matrix value =   8 ; sum of values =     8
   indices (  3,  1 ), matrix value =  17 ; sum of values =    25
   indices ( 10,  0 ), matrix value =  36 ; sum of values =    61
   indices (  9,  2 ), matrix value =  40 ; sum of values =   101
   indices ( 11,  3 ), matrix value =  52 ; sum of values =   153
   indices (  0,  9 ), matrix value =     ; sum of values =   153
   indices (  1,  8 ), matrix value =     ; sum of values =   153
   indices (  2, 10 ), matrix value =     ; sum of values =   153
   indices (  4,  5 ), matrix value =     ; sum of values =   153
   indices (  5, 11 ), matrix value =     ; sum of values =   153
   indices (  6,  7 ), matrix value =     ; sum of values =   153
   indices (  8,  6 ), matrix value =     ; sum of values =   153
  
 Example 2: Maximize the total benefit.
 
 my $N = 10;
 my $M = 10;
 my $r = 100;
 
 my @input_matrix;
 for my $i ( 0 .. $N - 1 ){
   my @weight_function;		   
   for my $j ( 0 .. $M - 1 ){
	  my $weight = sprintf( "%.0f", rand($r) );
      push @weight_function, $weight;
   }
   push @input_matrix, \@weight_function;
 }
 
 Alternatively, we can define the matrix with its elements:

 my @input_matrix = ( 
 [  84,  94,  75,  56,  66,  95,  39,  53,  73,   4 ],
 [  76,  71,  56,  49,  29,   1,  40,  40,  72,  72 ],
 [  85, 100,  71,  23,  47,  18,  82,  70,  30,  71 ],
 [   2,  95,  71,  89,  73,  73,  48,  52,  90,  51 ],
 [  65,  28,  77,  73,  24,  28,  75,  48,   8,  81 ],
 [  25,  27,  35,  89,  98,  10,  99,   3,  27,   4 ],
 [  58,  15,  99,  37,  92,  55,  52,  82,  73,  96 ],
 [  11,  75,   2,   1,  88,  43,   8,  28,  98,  20 ],
 [  52,  95,  10,  38,  41,  64,  20,  75,   1,  47 ],
 [  50,  80,  31,  90,  10,  83,  51,  55,  57,  40 ]
 );

 my ( $optimal, $assignment_ref, $output_index_ref ) = auction( matrix_ref => \@input_matrix, maximize_total_benefit => 1, verbose => 3 );

 Objective: to Maximize the total benefit
 Number of left nodes: 10
 Number of right nodes: 10
 Number of edges: 100

 Solution:
 Optimal assignment: sum of values = 893
 Feasible assignment condition: stepsize = 0.09091 < 1/10 = 0.1
 Number of iterations: 27

 row index    = [  0   1   2   3   4   5   6   7   8   9 ]
 column index = [  5   0   1   8   9   6   2   4   7   3 ]
 matrix value = [ 95  76 100  90  81  99  99  88  75  90 ]

 original matrix 10 x 10 with solution:
 [  84    94    75    56    66    95**  39    53    73     4  ]
 [  76**  71    56    49    29     1    40    40    72    72  ]
 [  85   100**  71    23    47    18    82    70    30    71  ]
 [   2    95    71    89    73    73    48    52    90**  51  ]
 [  65    28    77    73    24    28    75    48     8    81**]
 [  25    27    35    89    98    10    99**   3    27     4  ]
 [  58    15    99**  37    92    55    52    82    73    96  ]
 [  11    75     2     1    88**  43     8    28    98    20  ]
 [  52    95    10    38    41    64    20    75**   1    47  ]
 [  50    80    31    90**  10    83    51    55    57    40  ]

 Pairs (in ascending order of matrix values):
   indices (  8,  7 ), matrix value =  75 ; sum of values =  75
   indices (  1,  0 ), matrix value =  76 ; sum of values = 151
   indices (  4,  9 ), matrix value =  81 ; sum of values = 232
   indices (  7,  4 ), matrix value =  88 ; sum of values = 320
   indices (  3,  8 ), matrix value =  90 ; sum of values = 410
   indices (  9,  3 ), matrix value =  90 ; sum of values = 500
   indices (  0,  5 ), matrix value =  95 ; sum of values = 595
   indices (  5,  6 ), matrix value =  99 ; sum of values = 694
   indices (  6,  2 ), matrix value =  99 ; sum of values = 793
   indices (  2,  1 ), matrix value = 100 ; sum of values = 893

 Common use of the solution:
   
 foreach my $i ( sort { $a <=> $b } keys %{$assignment_ref} ){     
   my $j = $assignment_ref->{$i};   
   ...
 }

 my $sum = 0;
 for my $i ( 0 .. $#{$output_index_ref} ){
	my $j = $output_index_ref->[$i];   
	my $value = $input_matrix[$i]->[$j];
	$sum += $value if (defined $value);
	
	$value = defined $value ? sprintf( "%6s", $value ) : ' ' x 6 ; # %6s  
	printf " Auction Algorithm, output index --> \$i = %3d ; \$j = %3d ; \$value = $value ; \$sum = %8s \n", $i, $j, $sum;
 }
 
OPTIONS
 
 matrix_ref => \@input_matrix,   reference to array: matrix N x M.
 maximize_total_benefit => 0,    0: minimize the total benefit ; 1: maximize the total benefit.
 inicial_stepsize       => 1,    auction algorithm terminates with a feasible assignment if the problem data are integer and stepsize < 1/min(N,M).
 inicial_price          => 0,			
 verbose                => 3,    print messages on the screen, level of verbosity, 0: quiet; 1, 2, 3, 4, 5, 8, 10: debug information.

EXPORT

    "auction" function by default.

INPUT

    The input matrix should be in a two dimensional array (array of array) 
	and the 'auction' subroutine expects a reference to this array.

OUTPUT

    The $output_index_ref is the reference to the output_index array.
	The $assignment_ref  is the reference to the assignment hash.
	The $optimal is the total benefit which can be a minimum or maximum value.
	

SEE ALSO
  
	1. Network Optimization: Continuous and Discrete Models (1998).
	   Dimitri P. Bertsekas
	   http://web.mit.edu/dimitrib/www/netbook_Full_Book.pdf
	   
	2. Towards auction algorithms for large dense assignment problems (2008).
	   Libor Bus and Pavel Tvrdik
	   https://pdfs.semanticscholar.org/b759/b8fb205df73c810b483b5be2b1ded62309b4.pdf
	
	3. https://github.com/EvanOman/AuctionAlgorithmCPP/blob/master/auction.cpp
	   This Perl algorithm started from this C++ implementation.
	      
	4. https://en.wikipedia.org/wiki/Assignment_problem
	
	5. https://en.wikipedia.org/wiki/Auction_algorithm


AUTHOR

    Claudio Fernandes de Souza Rodrigues
	May 2, 2018
	Sao Paulo, Brasil
	claudiofsr@yahoo.com
	
COPYRIGHT AND LICENSE

 Copyright (c) 2018 Claudio Fernandes de Souza Rodrigues.  All rights reserved.

 This program is free software; you can redistribute it and/or modify
 it under the same terms as Perl itself.