Certifying Unsatisfiability of Random 2k-SAT Formulas Using Approximation Techniques

It is known that random k-SAT formulas with at least (2 · ln 2) ·n random clauses are unsatisfiable with high probability. This result is simply obtained by bounding the expected number of satisfying assignments of a random k-SAT instance by an expression tending to 0 when n, the number of variables tends to infinity. This argument does not give us an efficient algorithm certifying the unsatisfiability of a given random instance. For even k it is known that random k-SAT instances with at least Poly(log n) · n clauses can be efficiently certified as unsatisfiable. For k = 3 we need at least n + ε random clauses. In case of even k we improve the aforementioned results in two ways. There exists a constant C such that 4-SAT instances with at least C ·n clauses can be efficiently certified as unsatisfiable. Moreover, we give a satisfiability algorithm which runs in expected polynomial time over all k-SAT instances with C · n clauses. Our proofs are based on the direct application of known approximation algorithms on the one hand, and on a recent estimate of the θ-function for random graphs with a linear number of edges, on the other hand.