On the critical exponents of random k-SAT

There has been much recent interest in the satisfiability of random Boolean formulas. A random k-SAT formula is the conjunction of m random clauses, each of which is the disjunction of k literals (a variable or its negation). It is known that when the number of variables n is large, there is a sharp transition from satisfiability to unsatisfiability; in the case of 2-SAT this happens when m/n --> 1, for 3-SAT the critical ratio is thought to be m/n ~ 4.2. The sharpness of this transition is characterized by a critical exponent, sometimes called \nu=\nu_k (the smaller the value of \nu the sharper the transition). Experiments have suggested that \nu_3 = 1.5+-0.1, \nu_4 = 1.25+-0.05, \nu_5=1.1+-0.05, \nu_6 = 1.05+-0.05, and heuristics have suggested that \nu_k --> 1 as k --> infinity. We give here a simple proof that each of these exponents is at least 2 (provided the exponent is well-defined). This result holds for each of the three standard ensembles of random k-SAT formulas: m clauses selected uniformly at random without replacement, m clauses selected uniformly at random with replacement, and each clause selected with probability p independent of the other clauses. We also obtain similar results for q-colorability and the appearance of a q-core in a random graph.Comment: 11 pages. v2 has revised introduction and updated reference

Finding cores of random 2-SAT formulae via Poisson cloning

For the random 2-SAT formula $F(n,p)$, let $F_C (n,p)$ be the formula left after the pure literal algorithm applied to $F(n,p)$ stops. Using the recently developed Poisson cloning model together with the cut-off line algorithm (COLA), we completely analyze the structure of $F_{C} (n,p)$. In particular, it is shown that, for \gl:= p(2n-1) = 1+\gs with \gs\gg n^{-1/3}, the core of $F(n,p)$ has \thl^2 n +O((\thl n)^{1/2}) variables and \thl^2 \gl n+O((\thl n))^{1/2} clauses, with high probability, where \thl is the larger solution of the equation \th- (1-e^{-\thl \gl})=0. We also estimate the probability of $F(n,p)$ being satisfiable to obtain \pr[ F_2(n, \sfrac{\gl}{2n-1}) is satisfiable ] = \caseth{1-\frac{1+o(1)}{16\gs^3 n}}{if $\gl= 1-\gs$ with $\gs\gg n^{-1/3}$}{}{}{e^{-\Theta(\gs^3n)}}{if $\gl=1+\gs$ with $\gs\gg n^{-1/3}$,} where $o(1)$ goes to 0 as \gs goes to 0. This improves the bounds of Bollob\'as et al. \cite{BBCKW}

The unsatisfiability threshold revisited www.elsevier.com/locate/dam

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. © 2006 Elsevier B.V. All rights reserved