MAX-3SAT
problem in computer science
MAX-3SAT is a problem in the computational complexity subfield of computer science. It generalises the Boolean satisfiability problem (SAT) which is a decision problem considered in complexity theory. It is defined as:
Given a 3-CNF formula Φ (i.e. with at most 3 variables per clause), find an assignment that satisfies the largest number of clauses.
MAX-3SAT is a canonical complete problem for the complexity class MAXSNP (shown complete in Papadimitriou pg. 314).
Approximability
The decision version of MAX-3SAT is NP-complete. Therefore, a polynomial-time solution can only be achieved if P = NP. An approximation within a factor of 2 can be achieved with this simple algorithm, however:
Output the solution in which most clauses are satisfied, when either all variables = TRUE or all variables = FALSE.
Every clause is satisfied by one of the two solutions, therefore one solution satisfies at least half of the clauses.
The Karloff-Zwick algorithm runs in polynomial-time and satisfies ≥ 7/8 of the clauses. While this algorithm is randomized, it can be derandomized using, e.g., the techniques from to yield a deterministic (polynomial-time) algorithm with the same approximation guarantees.
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