60 research outputs found
On the Satisfiability Threshold and Clustering of Solutions of Random 3-SAT Formulas
We study the structure of satisfying assignments of a random 3-SAT formula.
In particular, we show that a random formula of density 4.453 or higher almost
surely has no non-trivial "core" assignments. Core assignments are certain
partial assignments that can be extended to satisfying assignments, and have
been studied recently in connection with the Survey Propagation heuristic for
random SAT. Their existence implies the presence of clusters of solutions, and
they have been shown to exist with high probability below the satisfiability
threshold for k-SAT with k>8, by Achlioptas and Ricci-Tersenghi, STOC 2006. Our
result implies that either this does not hold for 3-SAT or the threshold
density for satisfiability in 3-SAT lies below 4.453.
The main technical tool that we use is a novel simple application of the
first moment method
The Scaling Window of the 2-SAT Transition
We consider the random 2-satisfiability problem, in which each instance is a
formula that is the conjunction of m clauses of the form (x or y), chosen
uniformly at random from among all 2-clauses on n Boolean variables and their
negations. As m and n tend to infinity in the ratio m/n --> alpha, the problem
is known to have a phase transition at alpha_c = 1, below which the probability
that the formula is satisfiable tends to one and above which it tends to zero.
We determine the finite-size scaling about this transition, namely the scaling
of the maximal window W(n,delta) = (alpha_-(n,delta),alpha_+(n,delta)) such
that the probability of satisfiability is greater than 1-delta for alpha <
alpha_- and is less than delta for alpha > alpha_+. We show that
W(n,delta)=(1-Theta(n^{-1/3}),1+Theta(n^{-1/3})), where the constants implicit
in Theta depend on delta. We also determine the rates at which the probability
of satisfiability approaches one and zero at the boundaries of the window.
Namely, for m=(1+epsilon)n, where epsilon may depend on n as long as |epsilon|
is sufficiently small and |epsilon|*n^(1/3) is sufficiently large, we show that
the probability of satisfiability decays like exp(-Theta(n*epsilon^3)) above
the window, and goes to one like 1-Theta(1/(n*|epsilon|^3)) below the window.
We prove these results by defining an order parameter for the transition and
establishing its scaling behavior in n both inside and outside the window.
Using this order parameter, we prove that the 2-SAT phase transition is
continuous with an order parameter critical exponent of 1. We also determine
the values of two other critical exponents, showing that the exponents of 2-SAT
are identical to those of the random graph.Comment: 57 pages. This version updates some reference
A new upper bound for 3-SAT
We show that a randomly chosen 3-CNF formula over n variables with
clauses-to-variables ratio at least 4.4898 is, as n grows large, asymptotically
almost surely unsatisfiable. The previous best such bound, due to Dubois in
1999, was 4.506. The first such bound, independently discovered by many groups
of researchers since 1983, was 5.19. Several decreasing values between 5.19 and
4.506 were published in the years between. The probabilistic techniques we use
for the proof are, we believe, of independent interest.Comment: 20 page
Finding cores of random 2-SAT formulae via Poisson cloning
For the random 2-SAT formula , let be the formula left
after the pure literal algorithm applied to stops. Using the recently
developed Poisson cloning model together with the cut-off line algorithm
(COLA), we completely analyze the structure of . In particular, it
is shown that, for \gl:= p(2n-1) = 1+\gs with \gs\gg n^{-1/3}, the core of
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
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 goes to 0 as \gs goes to 0. This improves the
bounds of Bollob\'as et al. \cite{BBCKW}
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