lecture22 - CS 473 Algorithms Chandra Chekuri...

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Unformatted text preview: CS 473: Algorithms Chandra Chekuri [email protected] 3228 Siebel Center University of Illinois, Urbana-Champaign Fall 2009 Chekuri CS473ug Complementation Self Reduction Part I Complementation and Self-Reduction Chekuri CS473ug Complementation Self Reduction Recap Motivation co- NP Definition Relationship between P , NP and co- NP The class P A language L (equivalently decision problem) is in the class P if there is a polynomial time algorithm A for deciding L ; that is given a string x , A correctly decides if x ∈ L and running time of A on x is polynomial in | x | , the length of x . Chekuri CS473ug Complementation Self Reduction Recap Motivation co- NP Definition Relationship between P , NP and co- NP The class NP Two equivalent definitions: Language L is in NP if there is a non-deterministic polynomial time algorithm A (Turing Machine) that decides L . For x ∈ L , A has some non-deterministic choice of moves that will make A accept x For x 6∈ L , no choice of moves will make A accept x L has an efficient certifier C ( · , · ). C is a polynomial time deterministic algorithm For x ∈ L there is a string y (proof) of length polynomial in | x | such that C ( x , y ) accepts For x 6∈ L , no string y will make C ( x , y ) accept Chekuri CS473ug Complementation Self Reduction Recap Motivation co- NP Definition Relationship between P , NP and co- NP Complementation Definition Given a decision problem X , its complement ¯ X is the collection of all instances s such that s 6∈ X Equivalently, in terms of languages: Definition Given a language L over alphabet Σ, its complement ¯ L is the language Σ *- L . Chekuri CS473ug Complementation Self Reduction Recap Motivation co- NP Definition Relationship between P , NP and co- NP Examples PRIME = { n | n is an integer and n is prime } ¯ PRIME = { n | n is an integer and n is not a prime } ¯ PRIME = COMPOSITE SAT = { ϕ | ϕ is a SAT formula and ϕ is satisfiable } ¯ SAT = { ϕ | ϕ is a SAT formula and ϕ is not satisfiable } ¯ SAT = UnSAT Chekuri CS473ug Complementation Self Reduction Recap Motivation co- NP Definition Relationship between P , NP and co- NP Examples PRIME = { n | n is an integer and n is prime } ¯ PRIME = { n | n is an integer and n is not a prime } ¯ PRIME = COMPOSITE SAT = { ϕ | ϕ is a SAT formula and ϕ is satisfiable } ¯ SAT = { ϕ | ϕ is a SAT formula and ϕ is not satisfiable } ¯ SAT = UnSAT Technicality: ¯ SAT also includes strings that do not encode any valid SAT formula. Typically we ignore those strings because they are not interesting. In all problems of interest, we assume that it is “easy” to check whether a given string is a valid instance or not....
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lecture22 - CS 473 Algorithms Chandra Chekuri...

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