{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

NP_Complete_Reductions

NP_Complete_Reductions - Fall 2006 Costas Busch RPI 1...

Info icon This preview shows pages 1–10. Sign up to view the full content.

View Full Document Right Arrow Icon
Fall 2006 Costas Busch - RPI 1 More NP-complete Problems
Image of page 1

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

View Full Document Right Arrow Icon
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)
Image of page 2
Fall 2006 Costas Busch - RPI 3 Using the previous theorem, we will prove that 2 problems  are NP-complete: Vertex-Cover Hamiltonian-Path
Image of page 3

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

View Full Document Right Arrow Icon
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:
Image of page 4
Fall 2006 Costas Busch - RPI 5 |S|=4 Example: Size of vertex-cover  is the number of nodes in the cover
Image of page 5

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

View Full Document Right Arrow Icon
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
Image of page 6
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)
Image of page 7

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

View Full Document Right Arrow Icon
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
Image of page 8
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 + =
Image of page 9

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

View Full Document Right Arrow Icon
Image of page 10
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern