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Module Version: 0.05

# NAME

Algorithm::SetCovering - Algorithms to solve the "set covering problem"

# SYNOPSIS

```    use Algorithm::SetCovering;

my \$alg = Algorithm::SetCovering->new(
columns => 4,
mode    => "greedy");

my @to_be_opened = (@ARGV || (1, 1, 1, 1));

my @set = \$alg->min_row_set(@to_be_opened);

print "To open (@to_be_opened), we need ",
scalar @set, " keys:\n";

for(@set) {
print "\$_: ", join('-', \$alg->row(\$_)), "\n";
}```

# DESCRIPTION

Consider having M keys and N locks. Every key opens one or more locks:

```         | lock1 lock2 lock3 lock4
-----+------------------------
key1 |   x           x
key2 |   x     x
key3 |   x     x     x
key4 |         x           x
key5 |               x     x```

Given an arbitrary set of locks you have to open (e.g. 2,3,4), the task is to find a minimal set of keys to accomplish this. In the example above, the set [key4, key5] fulfils that condition.

The underlying problem is called "set covering problem" and the corresponding decision problem is NP-complete.

## Methods

\$alg = Algorithm::SetCovering->new(columns => \$cols, [mode => \$mode]);

Create a new Algorithm::SetCovering object. The mandatory parameter `columns` needs to be set to the number of columns in the matrix (the number of locks in the introductory example).

`mode` is optional and selects an algorithm for finding the solution. The following values for `mode` are implemented:

"brute_force"

Will iterate over all permutations of keys. Only recommended for very small numbers of keys.

"greedy"

Greedy algorithm. Scales O(mn^2). Can't do much better for a NP-hard problem.

The default for `mode` is set to "greedy".

Add a new row to the matrix. In the example above, this adds one key and specifies which locks it is able to open.

`    \$alg->add_row(1,0,0,1);`

specifies that the new key can open locks #1 and #4.

The number of elements in @columns needs to match the previously defined number of columns.

\$alg->min_row_set(@columns_to_cover)

Determines a minimal set of keys to cover a given set of locks and returns an array of index numbers for those keys.

Defines which columns have to be covered by passing in an array with true values on element positions that need to be covered. For example,

`    my @idx_set = \$alg->min_row_set(1,1,0,1);`

specifies that all but the third column have to be covered and returns an array of index numbers into an array, defined previously (and implicitely) via successive add_row() commands.

If no set of keys can be found that satisfies the given requirement, an empty list is returned.

If you've forgotten which locks the key referred to by a certain index number can open, use the `rows()` method to find out:

`    my(@opens_locks) = \$alg->rows(\$idx_set[0]);`

will give back an array of 0's and 1's, basically returning the very parameters we've passed on to the add_row() command previously.

## Strategies

Currently, the module implements the Greedy algorithm and also (just for scientific purposes) a dumb brute force method, creating all possible combinations of keys, sorting them by the number of keys used (combinations with fewer keys have priority) and trying for each of them if it fits the requirement of opening a given number of locks.

This obviously won't scale beyond a really small number of keys (N), because the number of permutations will be 2**N-2.

The Greedy Algorithm, on the other hand scales with O(mn^2), with m being the number of keys and n being the number of locks.

## Limitations

Julien Gervais-Bird <j.bird@usherbrooke.ca> points out: The greedy algorithm does not always return the minimal set of keys. Consider this example:

```         | lock1 lock2 lock3 lock4 lock5 lock6
-----+------------------------------------
key1 |   x           x           x
key2 |   x     x
key3 |               x     x
key4 |                           x     x```

The minimal set of keys to open all the locks is (key2, key3, key4), however the greedy algorithm will return (key1,key2,key3,key4) because key1 opens more locks than any other key.

# AUTHOR

Mike Schilli, 2003, <m@perlmeister.com>

Thanks to the friendly guys on rec.puzzles, who provided me with valuable input to analyze the problem and explained the algorithm:

```    Craig <c_quest000@yahoo.com>
Robert Israel <israel@math.ubc.ca>
Patrick Hamlyn <path@multipro.n_ocomsp_am.au>```

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

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