Sep 12, 2003
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- SAT problems
        - one of the first problems shown to be NP-complete
        - symbols, clauses, and CNF
        - most CSPs can be formulated as SAT! How?

- Examples of CNF SAT problems
        - (x1 OR x2) AND (x2 OR x3)
                - has many solutions
        - (x1 OR x2 OR x3) AND
          (~x1 OR x2) AND
          (~x2 OR x3) AND
          (~x3 OR x1) AND
          (~x1 OR ~x2 OR ~x3)
                - has no solution

- k-CNF-SAT
        - can have any number of clauses
        - can have any number of symbols
        - but at most k symbols/clause (or sometimes
          exactly k)

- Goal of boolean satisfiability
        - see if it is solvable
        - or say no, if no assignment exists

- 2-CNF-SAT is in P, but
        - 3- (and greater) CNF-STAT is NP-complete
        - brief digression about complexity classes

- solving simple SAT problems
        - truth table approach: O(2^n)
        - unit propagation and trying out values
                - polynomial time for 2-CNF-SAT

- Formulating other problems as CNF-SAT
        - n-queens
        - graph coloring

- What is hard about 3-CNF-SAT problems
        - ratio of clauses to variables (C/N ratio)
        - easy to solve when C/N is small
        - easy to prove unsatisfiability when C/N is large
        - middle region is most dangerous
                - around [4.2, 4.3]

- GSAT: an iterative improvement algorithm
        - two loops, with inner loop
        - flipping variables in direction of greatest
          improvement
        - sound but not guaranteed to get an answer
        - currently a "record holder" algorithm