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NP_Complete_Reductions

# NP_Complete_Reductions - Fall 2006 Costas Busch RPI 1...

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Fall 2006 Costas Busch - RPI 1 More NP-complete Problems

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Fall 2006 Costas Busch - RPI 2 Theorem: If:  Language       is NP-complete      Language       is in NP          is polynomial time reducible to A A B B Then:       is NP-complete B (proven in previous class)
Fall 2006 Costas Busch - RPI 3 Using the previous theorem, we will prove that 2 problems  are NP-complete: Vertex-Cover Hamiltonian-Path

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Fall 2006 Costas Busch - RPI 4 Vertex cover of a graph  is a subset of nodes      such that every edge  in the graph touches one node in  Vertex Cover S S = red nodes Example:
Fall 2006 Costas Busch - RPI 5 |S|=4 Example: Size of vertex-cover  is the number of nodes in the cover

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Fall 2006 Costas Busch - RPI 6 graph      contains a vertex cover  of size      } VERTEX-COVER  = {         :  k G , G k Corresponding language: Example: G COVER - VERTEX 4 , G
Fall 2006 Costas Busch - RPI 7 Theorem: 1.   VERTEX-COVER  is in NP 2.  We will reduce in polynomial time     3CNF-SAT  to  VERTEX-COVER Can be easily proven VERTEX-COVER  is NP-complete Proof: (NP-complete)

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Fall 2006 Costas Busch - RPI 8 Let      be a 3CNF formula with      variables  and    clauses  ϕ m l Example: ) ( ) ( ) ( 4 3 1 4 2 1 3 2 1 x x x x x x x x x = ϕ 4 = m 3 = l Clause 1 Clause 2 Clause 3
Fall 2006 Costas Busch - RPI 9 Formula       can be converted  to a graph       such that: ϕ G ϕ is satisfied if and only if G Contains a vertex cover of size l m k 2 + =

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