lec19-handout.pdf - Recap CMPSCI 311 Introduction to...

This preview shows 1 out of 2 pages.

CMPSCI 311: Introduction to Algorithms Lecture 19: Reductions and Intractability Akshay Krishnamurthy University of Massachusetts Last Compiled: April 18, 2018 Recap Reductions. Y P X if can solve Y in poly-time with algorithm for X . New problems. IndependentSet , VertexCover , SetCover , SAT , 3-SAT . Results. 3-SAT P IS P VC P SC VC P IS Reduction #3: Satisfiability Let X = { x 1 , . . . , x n } be boolean variables A term or literal is x i or ¬ x i . A clause is or of several terms ( t 1 t 2 . . . t ) . A formula is and of several clauses An assignment φ : X → { 0 , 1 } gives T/F to each variable. φ satisfies formula if all clauses evaluate to True. Example. ( x 1 ∨ ¬ x 2 ) ( x 1 x 4 ∨ ¬ x 3 ) ( ¬ x 1 x 4 ) ( x 3 x 2 ) Reduction #3: Satisfiability SAT – Given boolean formula C 1 C 2 . . . C m over variables X = { x 1 , . . . , x n } , does there exist a satisfying assignment? 3-SAT – Given boolean formula C 1 C 2 . . . C m over variables X = { x 1 , . . . , x n } where each C i has three literals, does there exist a satisfying assignment? Theorem. 3-SAT P IndependentSet . Reduction ( x 1 x 2 ∨ ¬ x 3 ) ( ¬ x 1 ∨ ¬ x 2 ∨ ¬ x 3 ) Associate nodes in graph with literals ( 2 per variable). Associate 3 nodes per clause in a gadget . If φ ( x i ) = 1 in assignment, then cannot select some nodes. x 1 ¬ x 1 x 2 ¬ x 2 x 3 ¬ x 3 x 1 x 2 ¬ x 3 ¬ x 1 ¬ x 2 ¬ x 3 Formally Given { x 1 , . . . , x n } and clauses C 1 , . . . , C m .
Image of page 1

Subscribe to view the full document.

Image of page 2
You've reached the end of this preview.
  • Fall '09
  • Computational complexity theory, Boolean satisfiability problem

{[ 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