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

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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
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?
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
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
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 even or odd “wings” of the gadget.
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• Fall '07
• staff
• Computational complexity theory, NP-complete problems, Knapsack problem, NP-complete, Boolean satisfiability problem

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

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