Theorem 3 coloring is NP complete COMP 6651 Fall 2013 Dr B Jaumard 49

Theorem 3 coloring is np complete comp 6651 fall 2013

This preview shows page 49 - 55 out of 55 pages.

Theorem . . . . . 3-coloring is NP-complete COMP 6651 / Fall 2013 , Dr. B. Jaumard 49
Image of page 49
. . . . . . . . . . . . . Definitions . . . . . . . . . Problem Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . NP-Completeness Proofs . Readings and References 3-Coloring (2/6) Given 3-SAT instance Φ , we construct an instance of 3-COLOR that is 3-colorable iff Φ is satisfiable. Construction (i) For each literal, create a node. (ii) Create 3 new nodes T, F, B; connect them in a triangle, and connect each literal to B. (iii) Connect each literal to its negation. (iv) For each clause, add gadget of 6 nodes and 13 edges. COMP 6651 / Fall 2013 , Dr. B. Jaumard 50
Image of page 50
. . . . . . . . . . . . . Definitions . . . . . . . . . Problem Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . NP-Completeness Proofs . Readings and References 3-Coloring (3/6) Suppose graph is 3-colorable. Consider assignment that sets all T literals to true. (ii) ensures each literal is T or F. (iii) ensures a literal and its negation are opposites. COMP 6651 / Fall 2013 , Dr. B. Jaumard 51
Image of page 51
. . . . . . . . . . . . . Definitions . . . . . . . . . Problem Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . NP-Completeness Proofs . Readings and References 3-Coloring (4/6) Suppose graph is 3-colorable. Consider assignment that sets all T literals to true. (ii) ensures each literal is T or F. (iii) ensures a literal and its negation are opposites. (iv) ensures at least one literal in each clause is T. COMP 6651 / Fall 2013 , Dr. B. Jaumard 52
Image of page 52
. . . . . . . . . . . . . Definitions . . . . . . . . . Problem Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . NP-Completeness Proofs . Readings and References 3-Coloring (5/6) Suppose graph is 3-colorable. Consider assignment that sets all T literals to true. (ii) ensures each literal is T or F. (iii) ensures a literal and its negation are opposites. (iv) ensures at least one literal in each clause is T. COMP 6651 / Fall 2013 , Dr. B. Jaumard 53
Image of page 53
. . . . . . . . . . . . . Definitions . . . . . . . . . Problem Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . NP-Completeness Proofs . Readings and References 3-Coloring (6/6) Suppose 3-SAT formula Φ is satisfiable. Color all true literals T. Color node below green node F, and node below that B. Color remaining middle row nodes B. Color remaining bottom nodes T or F as forced. COMP 6651 / Fall 2013 , Dr. B. Jaumard 54
Image of page 54
. . . . . . . . . . . . . Definitions . . . . . . . . . Problem Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . NP-Completeness Proofs . Readings and References . . . 1 Cormen, T.H., C.E. Leiserson, R.L. Rivest, and C. Stein, Introduction to Algorithms , Second Edition, McGraw Hill, 2001. . . . 2 Garey, M.R., and D.S. Johnson, Computers and Intractability- A Guide to the Theory of NP-Completeness , Freeman, 1979. COMP 6651 / Fall 2013 , Dr. B. Jaumard 55
Image of page 55

You've reached the end of your free preview.

Want to read all 55 pages?

  • Fall '09
  • Computational complexity theory, NP-complete problems, NP-complete, Professor B. Jaumard

  • Left Quote Icon

    Student Picture

  • Left Quote Icon

    Student Picture

  • Left Quote Icon

    Student Picture

Stuck? We have tutors online 24/7 who can help you get unstuck.
A+ icon
Ask Expert Tutors You can ask 0 bonus questions You can ask 5 questions (5 expire soon) You can ask 5 questions (will expire )
Answers in as fast as 15 minutes