Abstract | The problem of determining the unsatisfiability threshold for random 3-SAT formulas consists in determining the clause to variable
ratio that marks the experimentally observed abrupt change from almost surely satisfiable formulas to almost surely unsatisfiable. Up
to now, there have been rigorously established increasingly better lower and upper bounds to the actual threshold value. In this paper,
we consider the problem of bounding the threshold value from above using methods that, we believe, are of interest on their own
right. More specifically, we show how the method of local maximum satisfying truth assignments can be combined with results for
the occupancy problem in schemes of random allocation of balls into bins in order to achieve an upper bound for the unsatisfiability
threshold less than 4.571. In order to obtain this value, we establish a bound on the q-binomial coefficients (a generalization of the
binomial coefficients). No such bound was previously known, despite the extensive literature on q-binomial coefficients. Finally,
to prove our result we had to establish certain relations among the conditional probabilities of an event in various probabilistic
models for random formulas. It turned out that these relations were considerably harder to prove than the corresponding ones for
unconditional probabilities, which were previously known. |