######################################################################
    Algorithm::SetCovering 0.03
######################################################################

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

SYNOPSIS
        use Algorithm::SetCovering;

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

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

        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".

    $alg->add_row(@columns)
        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.

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>

COPYRIGHT AND LICENSE
    Copyright 2003 by Mike Schilli

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

