lec19-handout.pdf

# lec19-handout.pdf - Recap CMPSCI 311 Introduction to...

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CMPSCI 311: Introduction to Algorithms Lecture 19: Reductions and Intractability Akshay Krishnamurthy University of Massachusetts Last Compiled: April 18, 2018 Recap Reductions. Y P X if can solve Y in poly-time with algorithm for X . New problems. IndependentSet , VertexCover , SetCover , SAT , 3-SAT . Results. 3-SAT P IS P VC P SC VC P IS Reduction #3: Satisfiability Let X = { x 1 , . . . , x n } be boolean variables A term or literal is x i or ¬ x i . A clause is or of several terms ( t 1 t 2 . . . t ) . A formula is and of several clauses An assignment φ : X → { 0 , 1 } gives T/F to each variable. φ satisfies formula if all clauses evaluate to True. Example. ( x 1 ∨ ¬ x 2 ) ( x 1 x 4 ∨ ¬ x 3 ) ( ¬ x 1 x 4 ) ( x 3 x 2 ) Reduction #3: Satisfiability SAT – Given boolean formula C 1 C 2 . . . C m over variables X = { x 1 , . . . , x n } , does there exist a satisfying assignment? 3-SAT – Given boolean formula C 1 C 2 . . . C m over variables X = { x 1 , . . . , x n } where each C i has three literals, does there exist a satisfying assignment? Theorem. 3-SAT P IndependentSet . Reduction ( x 1 x 2 ∨ ¬ x 3 ) ( ¬ x 1 ∨ ¬ x 2 ∨ ¬ x 3 ) Associate nodes in graph with literals ( 2 per variable). Associate 3 nodes per clause in a gadget . If φ ( x i ) = 1 in assignment, then cannot select some nodes. x 1 ¬ x 1 x 2 ¬ x 2 x 3 ¬ x 3 x 1 x 2 ¬ x 3 ¬ x 1 ¬ x 2 ¬ x 3 Formally Given { x 1 , . . . , x n } and clauses C 1 , . . . , C m .

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• Fall '09
• Computational complexity theory, Boolean satisfiability problem

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