3-CNF-satisfiability in polynomial time

Abstract

For more than four decades, satisfiability of 2-CNF (2-SAT) is well-known to be solvable in polynomial (in fact linear) time, while satisfiability of 3-CNF formula (3-SAT) is NP-Complete. For last four decades, it was widely believed that 3-SAT does not have polynomial algorithm [6]. We fix this anomaly by first giving two polynomial algorithmic formulations for 2-SAT. Further, we introduce the notion of "context". For a clause of length 3, we generate 3 contexts. Suppose that we have a 3-CNF formula F with k clauses. We represent formula F by generating 3*k contexts, storing all of them, and analyzing them collectively. We construct one 2-SAT problem for analyzing each context. This approach results in the most surprising result that 3-SAT is also solvable in polynomial time. We demonstrate a simple bound of O(k3n3) set operations for our polynomial algorithm for 3-SAT.