NP_Complete_Reductions

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

Info iconThis preview shows pages 1–9. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Fall 2006 Costas Busch - RPI 1 More NP-complete Problems 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 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 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 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) 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...
View Full Document

This note was uploaded on 12/02/2011 for the course AR 107 taught by Professor Gracegraham during the Fall '11 term at Montgomery College.

Page1 / 30

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

This preview shows document pages 1 - 9. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online