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]) ** 2SYNOPSIS
### --- simple and direct application --- ###
### --- start --- ###
#!/usr/bin/perl
use strict;
use warnings FATAL => 'all';
use diagnostics;
use Algorithm::Bertsekas qw(auction); # To install this modulus: 'cpan Algorithm::Bertsekas' or 'ppm install Algorithm-Bertsekas'
my @array1; my @array2;
my @input_matrix;
my $N = 10;
my $M = 10;
my $range = 1000;
for my $i (1..$N) {
push @array1, sprintf( "%.0f", rand($range) );
}
for my $i (1..$M) {
push @array2, sprintf( "%.0f", rand($range) );
}
print "\n \@array1 = ( ";
for my $value (@array1) { printf("%4.0f ", $value); }
print ")\n";
print " \@array2 = ( ";
for my $value (@array2) { printf("%4.0f ", $value); }
print ")\n";
for my $i ( 0 .. $#array1 ){
my @weight_function;
for my $j ( 0 .. $#array2 ){
#my $weight = sprintf( "%.0f", rand($range) );
my $weight = abs ($array1[$i] - $array2[$j]);
push @weight_function, $weight;
}
push @input_matrix, \@weight_function;
}
print "\n The Nearest Neighbors and the Matrix of the weight function f(i,j) between each element of the two vectors \@array1 and \@array2.";
print "\n The weight function chosen can be the modulus of the difference between two real numbers: f(i,j) = abs (\$array1[i] - \$array2[j]). \n\n \@input_matrix = \n\n ";
print " " x 7;
printf("%4.0f ", $_ ) for @array2;
print "\n\n";
for my $i ( 0 .. $#input_matrix ) {
printf(" %4.0f [ ", $array1[$i] );
for my $j ( 0 .. $#{$input_matrix[$i]} ) {
printf("%4.0f ", $input_matrix[$i]->[$j] );
}
print "]\n";
}
my ( $optimal, $assignment_ref, $output_index_ref ) = auction( matrix_ref => \@input_matrix, maximize_total_benefit => 0, verbose => 10 );
print "\n";
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, (row, column) indexes --> \$i = %3d ; \$j = %3d ; \$value = $value ; \$sum = %8s \n", $i, $j, $sum;
}
### --- final --- ###
### --- simple and direct application --- ###
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):
indexes ( 7, 4 ), matrix value = 8 ; sum of values = 8
indexes ( 3, 1 ), matrix value = 17 ; sum of values = 25
indexes ( 10, 0 ), matrix value = 36 ; sum of values = 61
indexes ( 9, 2 ), matrix value = 40 ; sum of values = 101
indexes ( 11, 3 ), matrix value = 52 ; sum of values = 153
indexes ( 0, 9 ), matrix value = ; sum of values = 153
indexes ( 1, 8 ), matrix value = ; sum of values = 153
indexes ( 2, 10 ), matrix value = ; sum of values = 153
indexes ( 4, 5 ), matrix value = ; sum of values = 153
indexes ( 5, 11 ), matrix value = ; sum of values = 153
indexes ( 6, 7 ), matrix value = ; sum of values = 153
indexes ( 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):
indexes ( 8, 7 ), matrix value = 75 ; sum of values = 75
indexes ( 1, 0 ), matrix value = 76 ; sum of values = 151
indexes ( 4, 9 ), matrix value = 81 ; sum of values = 232
indexes ( 7, 4 ), matrix value = 88 ; sum of values = 320
indexes ( 3, 8 ), matrix value = 90 ; sum of values = 410
indexes ( 9, 3 ), matrix value = 90 ; sum of values = 500
indexes ( 0, 5 ), matrix value = 95 ; sum of values = 595
indexes ( 5, 6 ), matrix value = 99 ; sum of values = 694
indexes ( 6, 2 ), matrix value = 99 ; sum of values = 793
indexes ( 2, 1 ), matrix value = 100 ; sum of values = 893
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, 9, 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_algorithmAUTHOR
Claudio Fernandes de Souza Rodrigues
May 21, 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.1 POD Error
The following errors were encountered while parsing the POD:
- Around line 792:
Non-ASCII character seen before =encoding in 'N×M'. Assuming CP1252