Howdy Monks,

I have a set of objects, each of which has a collection of text attributes. I am trying to find groups of those objects with the most, and largest, sets of attributes in common. If I require all objects in a grouping to have all the attributes for it to count as a set, then clearly I'm just taking the intersection of the attribute sets, which is fairly simple. But I want to know about "partial sets" as well - I think an example will serve me best:

"apple" => ("red", "round", "plant", "fruit")
"orange" => ("orange", "round", "plant", "fruit")
"pumpkin" => ("orange", "round", "plant", "vegetable")
"ball" => ("red", "round", "toy")
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EDIT: Updated this, as my original example was unclear.

For a group size of 3, I check the sets of each permutation of 3 objects:

(apple, orange, pumpkin):
  orange: 2
  round: 3
  plant: 3
  fruit: 2
(apple, orange, ball):
  red: 2
  round: 3
  plant: 2
  fruit: 2
(apple, pumpkin, ball):
  red: 2
  round: 3
  plant:2
(orange, pumpkin, ball):
  orange: 2
  round: 3
  plant: 2
[download]

(Ignoring sets of size 1 as uninteresting...) Again, this is not that hard to calculate for a given group. But what I'm trying to do is _find_ the groups with the most/largest sets (I'm provisionally using a scoring system which I _think_ is mostly irrelevant to my problem...) Brute force checking the score of all the combinations of 4 from my collection of 500-odd objects leaves me checking over fifty billion combinations; in addition to offending my sensibilities, this is kinda slow. So I'm hoping your Monkishnesses can help come up with an algorithm / search strategy for constructing sets to score that are more likely to be "interesting". Maybe something like:

For each pairing of two objects, find the intersection of the two attribute sets.

_For each pair intersection, ordered by the size of the intersect descending and above some minimum size, calculate the intersect with each other object

__For each triplet intersection, ordered by the size of the intersect descending and above some minimum size, calculate the score for the set of objects

(Just thought of that then. I'll have to go try it, though it is pretty fuzzy still.) The idea being that each time you cut off a block of intersections for being below a minimum size, you chop out a huge chunk of the search space. But since I now am not doing a complete search of the combination space, don't I run the risk of coming at the same combination from different directions? - like measuring both of "apple", "pumpkin", "ball" and "ball", "apple", "pumpkin - and doing the work twice.

In any case, I appreciate any ideas, and apologize in advance if this is too much of an algorithm question that isn't related enough to perl specifically.