Group/Cluster Object List Based on multiple discrete attributes

I have a list of objects with attributes eg Color, Length, Width and importantly Price. I am ultimately trying to partition the list into the largest subsets so that each subset member has the same price and every permutation of color, length, width is present in the 开发者_JS百科subset.

In other words I want to be able to divide the item set to discover subsets so that each set can be described like:

• all objects that are red, blue, or green and either 10 or 11 inches long as well as 4 inches high cost $5
• All objects that are red or green and 12 inches long and 4 inches high are $6
• A blue 12X4 object is $6.50
• etc. etc.

with the goal of having the subsets contain the most items possible.

I've been wrestling with the problem for a while now, and I think the answer (or a close approximation) might come from clustering techniques. However, I've been hard pressed to come up with a distance matrix that will provide the results I am looking for. Does anyone have any insight into how to solve this problem.

Better Explanation

Given a list of objects similar to the ones above group them into the smallest number of complete subsets by price.

Well, step one is to partition the whole thing on price.

From there it sounds like you want a completeness sample over the other traits, not clustering. If you want "every permutation of color, etc." that isn't clustering, in any ways that I understand! It sounds like anti-clustering!

In any case, I find your description of your end goal quite confusing.