Locally satisfiable formulas

• Authors: Daniel Král
• Venue: SODA '04 Proceedings of the fifteenth annual ACM-SIAM symposium on Discrete algorithms
• Year: 2004

A CNF formula ψ is k-satisfiable if each k clauses of ψ can be satisfied simultaneously. Let πk be the largest real number such that for each k-satisfiable formula with variables xi, there are probabilities pi with the following property: If each variable xi is chosen randomly and independently to be true with the probability pi, then each clause of is satisfied with the probability at least πk.We determine the numbers πk and design a linear-time algorithm which given a formula either outputs that ψ is not k-satisfiable or finds probabilities pi such that each clause ψ of is satisfied with the probability at least πk. Our approach yields a robust linear-time deterministic algorithm which finds for a k-satisfiable formula a truth assignment satisfying at least the fraction of πk of the clauses.A related parameter is rk which is the largest ratio such that for each k-satisfiable CNF formula with m clauses, there is a truth assignment which satisfies at least rkm clauses. It was known that πk = rk for k = 1, 2, 3. We compute the ratio r4 and show π4 6 = r4. We also design a linear-time algorithm which finds a truth assignment satisfying at least the fraction r4 of the clauses for 4-satisfiable formulas.