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: Minimize the total benefit.
my @array1 = ( 22, 15, 98, 1 );
my @array2 = ( 72, 99, 29, 88, 12, 26, 41 );
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;
}
72 99 29 88 12 26 41
22 [ 50 77 7 66 10 4 19 ]
15 [ 57 84 14 73 3 11 26 ]
98 [ 26 1 69 10 86 72 57 ]
1 [ 71 98 28 87 11 25 40 ]
my ( $optimal, $assignement_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: 4
Number of right nodes: 7
Number of edges: 28
Solution:
Optimal assignment: sum of values = 30
Feasible assignment condition: stepsize = 0.2 < 1/4 = 0.25
Number of iterations: 15
row index = [ 0 1 2 3 4 5 6 ]
column index = [ 2 5 1 4 0 6 3 ]
matrix value = [ 7 11 1 11 ]
modified matrix 4 x 7:
[ 48 21 91** 32 88 94 79 ]
[ 41 14 84 25 95 87** 72 ]
[ 72 97** 29 88 12 26 41 ]
[ 27 0 70 11 87** 73 58 ]
original matrix 4 x 7 with solution:
[ 50 77 7** 66 10 4 19 ]
[ 57 84 14 73 3 11** 26 ]
[ 26 1** 69 10 86 72 57 ]
[ 71 98 28 87 11** 25 40 ]
Pairs (in ascending order of matrix values):
indices ( 2, 1 ), matrix value = 1 ; sum of values = 1
indices ( 0, 2 ), matrix value = 7 ; sum of values = 8
indices ( 1, 5 ), matrix value = 11 ; sum of values = 19
indices ( 3, 4 ), matrix value = 11 ; sum of values = 30
indices ( 4, 0 ), matrix value = ; sum of values = 30
indices ( 5, 6 ), matrix value = ; sum of values = 30
indices ( 6, 3 ), matrix value = ; sum of values = 30
Example 2: Maximize the total benefit.
Alternatively, we can define the matrix with its elements:
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;
}
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, $assignement_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 <=> $b } keys %{$assignement_ref} ){
my $j = $assignement_ref->{$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, 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 $assignement_ref is the reference to the assignement 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
March 31, 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.