3dm - CMSC 451: More NP-completeness Results Slides By:...

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CMSC 451: More NP-completeness Results Slides By: Carl Kingsford Department of Computer Science University of Maryland, College Park Based on Sect. 8.5,8.7,8.9 of Algorithm Design by Kleinberg & Tardos.
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Three-Dimensional Matching Three-Dimensional Matching
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Two-Dimensional Matching Recall ‘2-d matching’: Given sets X and Y , each with n elements, and a set E of pairs { x , y } , Question: is there a choice of pairs such that every element in X Y is paired with some other element? Usually, we thought of edges instead of pairs : { x , y } , but they are really the same thing.
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Three-Dimensional Matching X Y Z Given: Sets X , Y , Z , each of size n , and a set T X × Y × Z of order triplets. Question: is there a set of n triplets in T such that each element is contained in exactly one triplet?
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3DM Is NP-Complete Theorem Three-dimensional matching (aka 3DM) is NP-complete Proof. 3DM is in NP: a collection of n sets that cover every element exactly once is a certificate that can be checked in polynomial time. Reduction from 3-SAT. We show that: 3-SAT P 3DM In other words, if we could solve 3DM, we could solve 3-SAT.
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3-SAT P 3DM 3SAT instance: x 1 , . . . , x n be n boolean variables, and C 1 , . . . , C k clauses. We create a gadget for each variable x i : A i = { a i 1 , . . . , a i , 2 k } core B i = { a i 1 , . . . , a i , 2 k } tips t ij = ( a ij , a i , j +1 , b ij ) TF triples a 11 a 12 a 13 a 14 b 11 b 12 b 13 b 14 t 11 t 12 t 13 t 14
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Gadget Encodes True and False a 11 a 12 a 13 a 14 b 11 b 12 b 13 b 14 t 11 t 12 t 13 t 14
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Gadget Encodes True and False a 11 a 12 a 13 a 14 b 11 b 12 b 13 b 14 t 11 t 12 t 13 t 14
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Gadget Encodes True and False a 11 a 12 a 13 a 14 b 11 b 12 b 13 b 14 t 11 t 12 t 13 t 14
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How “choice” is encoded We can only either use the
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This note was uploaded on 01/13/2012 for the course CMSC 423 taught by Professor Staff during the Fall '07 term at Maryland.

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3dm - CMSC 451: More NP-completeness Results Slides By:...

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