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# Maximal Elements in a Lower Set

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Problem Detail:

I have a collection of objects, and a feasibility property for sets of objects which is slow to compute. If a set is feasible then so is any subset. For example, it could be whether the set of things will fit into a certain-sized packing crate.

I want to compute all feasible sets (equivalently - all maximal feasible sets), minimizing the number of evaluations of the feasibility property (worst-case or average). Does anyone know of any theoretical work on this?

Bonus question - what if the cost to compute the property is linear (or superlinear) in the size of the set being evaluated?

###### Answered By : Yuval Filmus

Your problem is equivalent to exact learning of monotone DNFs. This task is equivalent to learning all minterms of a monotone function, whereas you are interested in learning all maxterms of a monotone function, but the two problems are equivalent. (Look these terms up if you are not familiar with them.)

If the only queries you allow are membership queries (test whether a given set is feasible or not) then your problem is rather difficult. Theorem 2 in Angluin's Queries and Concept Learning shows that \$2^n-1\$ queries are required even if there are only \$2n\$ objects and \$n+1\$ minterms. See the paper for other models in which your problem can be solved efficiently.

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Question Source : http://cs.stackexchange.com/questions/64833

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