easy combinatorial algorithm, or not??

Hi all,

I'm stumped on finding an polytime algorithm for the following
seemingly easy subset problem. Does anyone have any information on this
problem
(identification/complexity/reduction/efficient solution, etc)?

Ideally, I'd like a polytime algorithm for this, or some concrete
indication that it's NP.

Given:
1) a set of sorted positive integers S = {s_1, ..., s_t} without
repetition.
2) a integer constant alpha, 1 <= alpha <= |S|
3) a multiset R of different (possibly all) alpha-subsets of S

Problem:
Given an arbitrary subset T of S s.t. |T| >= alpha, are all of T's
alpha-subsets contained in R?

Ideally, this would be done without enumeration of all of T's
alpha-subsets and without explicitly storing R, as that pushes us into
factorial time.

Example:
S = {0, 1, 2, 3, 4, 5, 6}
alpha = 3
R = { {0,1,2}, {0,1,3}, {0,1,4}, {2,4,5}, {4,5,6} }
T = {0, 1, 2, 3}
Question: Are all of T's 3-subsets contained in R?
Answer: No, because {1,2,3} and {0,2,3} are not contained in R.


Thanks for any info/help,

- Mayank

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