Abstract | One of the most challenging problems in probability and complexity theory is
to establish and determine the satisfiability threshold, or phase transition, for
random k-SAT instances: Boolean formulas consisting of clauses with exactly k
literals. As the previous part of the volume has explored, empirical observations
suggest that there exists a critical ratio of the number of clauses to the number
of variables, such that almost all randomly generated formulas with a higher
ratio are unsatisfiable while almost all randomly generated formulas with a lower
ratio are satisfiable. The statement that such a crossover point really exists is
called the satisfiability threshold conjecture. Experiments hint at such a direction,
but as far as theoretical work is concerned, progress has been difficult. In an
important advance, Friedgut [23] showed that the phase transition is a sharp one,
though without proving that it takes place at a “fixed” ratio for large formulas.
Otherwise, rigorous proofs have focused on providing successively better upper
and lower bounds for the value of the (conjectured) threshold. In this chapter, our |