• SAT
  • given an expression in terms of:
    • Variables
    • and, not, or
    • if, iff
    • ()
  • Is there a set of input for the variables that will produce a 1 (or true)?
  • ((w → x ) ∨ ¬((¬ w ↔ y) ∨ z)) ∧ ¬ x
    • w = 0, x=0, y = 1, z = 1
    • ((0 → 0 ) ∨ ¬((¬ 0 ↔ 1) ∨ 1)) ∧ ¬ 0
    • (1 ∨ 0) ∧ 1
    • 1
    • So this equation is satisfiable.
  • Again, a 2ⁿ algorithm will solve this.
  • But is it in NP-Complete?
    • SAT ∈ NP, fairly easy
    • CIRCUIT-SAT ≤_P SAT
      • The brute force approach to the reduction is not polynomial.
      • Show this.
      • Trick, use and of iff.
      • Show this.
• After this, CLRS go on to show quite a few reductions
  • 3-CNF-SAT: is an expression in the form (x₁ ∨ x₂ ∨ ¬ x₃) ∧ (x₂ ∨ ¬ x₄ ∨ x₅) ∧ ... satisfiable?
  • SAT ≤_P 3-CNF
  • A clique in a graph G is a set of completely connected vertexes.
  • Finding the maximum sized clique in a graph is the clique problem
  • CLIQUE: given a graph G does a clique of size k exist?
  • 3-CNF-SAT ≤_P CLIQUE
  • CLIQUE ≤ _P HAM-CYCLE
  • HAM-CYCLE ≤_P TSP ...
  • Each proof is unique.