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NP_Complete - NPcompleteLanguages f w k O(| w |...

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Fall 2006 Costas Busch - RPI 1 NP-complete Languages
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Fall 2006 Costas Busch - RPI 2 Polynomial Time Reductions Polynomial Computable function      :  f such that for any string       computes in polynomial time:    ) ( w f w There is a deterministic Turing machine ) | (| k w O M
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Fall 2006 Costas Busch - RPI 3 ) | (| | ) ( | k w O w f = since,        cannot use  more than                tape space in time  M ) | (| k w O ) | (| k w O Observation:
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Fall 2006 Costas Busch - RPI 4 Language  is polynomial time reducible to   language if there is a polynomial computable  function     such that: f B w f A w ) ( A B Definition:
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Fall 2006 Costas Busch - RPI 5 Suppose that      is polynomial reducible to     . If               then             .   Theorem: P B A B P A Proof: Machine        to decide      in polynomial time: A On input string      : w 1. Compute ) ( w f Let        be the machine that decides         B M 2. Run       on input  ) ( w f M in polynomial time 3. If              acccept  B w f ) ( w M
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Fall 2006 Costas Busch - RPI 6 Example of a polynomial-time reduction: We will reduce the  3CNF-satisfiability  problem to the CLIQUE   problem
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Fall 2006 Costas Busch - RPI 7 3CNF formula: ) ( ) ( ) ( ) ( 6 5 4 4 6 3 6 5 3 3 2 1 x x x x x x x x x x x x Each clause has three literals 3CNF-SAT  ={       :       is a satisfiable                                      3CNF formula} w w Language: literal clause
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Fall 2006 Costas Busch - RPI 8 5 -clique in graph CLIQUE  = {            :  graph                                         contains a    -clique} < k G , G k G Language:
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Fall 2006 Costas Busch - RPI 9 Theorem: 3CNF-SAT  is polynomial time  reducible to  CLIQUE Proof: give a polynomial time reduction of one problem to the other Transform formula to graph
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Fall 2006 Costas Busch - RPI 10 ) ( ) ( ) ( ) ( 4 3 2 3 2 1 4 2 1 4 2 1 x x x x x x x x x x x x Clause 2 Clause 1 Clause 3 1 x 2 x 1 x 2 x 4 x 1 x 2 x 3 x 2 x 4 x 4 x 3 x Transform formula to graph.  Example: Clause 4 Create Nodes:
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Fall 2006 Costas Busch - RPI 11 ) ( ) ( ) ( ) ( 4 3 2 3 2 1 4 2 1 4 2 1 x x x x x x x x x x x x 1 x 2 x 1 x 2 x 4 x 1 x 2 x 2 x 4 x 4 x 3 x 3 x Add link from a literal     to a literal in every other clause, except the complement ξ ξ
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Fall 2006 Costas Busch - RPI 12 ) ( ) ( ) ( ) ( 4 3 2 3 2 1 4 2 1 4 2 1 x x x x x x x x x x x x 1 x 2 x 1 x 2 x 4 x 1 x 2 x 3 x 2 x 4 x 4 x 3 x Resulting Graph
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Fall 2006 Costas Busch - RPI 13 1 0 0 1 4 3 2 1 = = = = x x x x 1 ) ( ) ( ) ( ) ( 4 3 2 3 2 1 4 2 1 4 2 1 =
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