# 3SAT2DM - 3DM Instance: A positive integer q, a set of 3q...

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3–DM Instance: A positive integer q, a set of 3q distinct objects partitioned into three sets of q objects W, X, and Y. And, a set of s "triples" of the form (a, b, c) where a W, b X, and c Y. Question: Does there exist a subset of exactly q triples containing all of the 3q objects? 3–SAT 3–DM 3–DM is clearly in NP. Consider an arbitrary instance of 3–SAT, U = {u 1 , u 2 , …, u m } and a set of m clauses C 1 , C 2 , …, C m . Each C i contains exactly 3 variables of the form u or u'. Construct W to be {u i [j], u' i [j] | for 1 ≤ i ≤ n and 1 ≤ j ≤ m}. Note: u 1 [1], u 1 [2], … , u 1 [n] all correspond to Boolean variable u i . while u' 1 [1], u' 1 [2], … , u' 1 [n] all correspond to the complement of u i . Later triples, corresponding to Boolean variables in, say, clause j, will force us to "select" a triple containing u i [j]. If u i also appears in clause k, we must also select the triple containing u i [k], etc. Then, of course, none of these triples can ever force us to select u'

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## This note was uploaded on 07/14/2011 for the course COT 4610 taught by Professor Dutton during the Fall '10 term at University of Central Florida.

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3SAT2DM - 3DM Instance: A positive integer q, a set of 3q...

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