Further NP Complete Proofs

• Clean up a few definitions.
  • A problem h is in NP-HARD if every problem in NP can be reduced to h in polynomial time.
    • This can include problems that are not in NP.
    • (Stolen from Wikipedia).
  • A problem c is in co-NP is the set of problems who's compliment is in the class NP.
    • We really don't need this.
• We have seen how CIRCUIT-SAT is NP-COMPLETE.
• It becomes much easier to show other problems are NP-COMPLETE now just by reducing CIRCUIT-SAT to them.
• A next logical step is SAT
  • Given a boolean formula
    • Composed of n variables $x_1, x_2, ... x_n$
    • M boolean operations (AND, OR, NOT, ...) with two inputs.
    • Parenthesis to change order.
    is there a input such that the formula produces a true (or 1)?
• SAT $\in $ NP
  • Clearly given an input, we can evaluate it in polynomial time.
• The proof that SAT is NP-COMPLETE is done by CIRCUIT_SAT $\lt_P$ SAT
  • While not quite a straight forward as you might think, this is not too bad.
  • The problem is keeping the construction of an equation from a circuit polynomial.
  • When the gates can have more than two inputs.
  • But it can be done.
• This continues
  • Stolen from Wikipedia again, but they took it from CLRS.
  • Or a larger graph on line
• Wikipedia has an article listing some problems.
  • And the list continues to grow.
  • Mario Brothers is harder than NP-Hard.