Here is an algorithm - certainly not the best - that would work: If you could load your dictionary into memory:

  my @words = <DICTIONARY>;
[download]

Then it is a simple process of taking your input letters, say art and doing:

  #! /usr/bin/perl -w
  use strict;

  my $input = 'art';

  # Read words from a file instead.
  my @words = ('tar', 'rat', 'fake', 'at', 'unknown');

  my $regex = qr{[$input]+};
  my @matches = grep /^$regex$/, @words;
  print "$_\n" for @matches;
[download]

Beware of: upper/lower case in input and in the dictionary, the massive amount of memory the words array will consume...