3 research outputs found
Analysing Survey Propagation Guided Decimationon Random Formulas
Let be a uniformly distributed random -SAT formula with
variables and clauses. For clauses/variables ratio the formula is satisfiable with high
probability. However, no efficient algorithm is known to provably find a
satisfying assignment beyond with a non-vanishing
probability. Non-rigorous statistical mechanics work on -CNF led to the
development of a new efficient "message passing algorithm" called \emph{Survey
Propagation Guided Decimation} [M\'ezard et al., Science 2002]. Experiments
conducted for suggest that the algorithm finds satisfying assignments
close to . However, in the present paper we prove that the
basic version of Survey Propagation Guided Decimation fails to solve random
-SAT formulas efficiently already for
with almost a factor below
.Comment: arXiv admin note: substantial text overlap with arXiv:1007.1328 by
other author
Counting Solutions to Random CNF Formulas
We give the first efficient algorithm to approximately count the number of
solutions in the random -SAT model when the density of the formula scales
exponentially with . The best previous counting algorithm was due to
Montanari and Shah and was based on the correlation decay method, which works
up to densities , the Gibbs uniqueness threshold
for the model. Instead, our algorithm harnesses a recent technique by Moitra to
work for random formulas. The main challenge in our setting is to account for
the presence of high-degree variables whose marginal distributions are hard to
control and which cause significant correlations within the formula
Improved Bounds for Sampling Solutions of Random CNF Formulas
Let be a random -CNF formula on variables and clauses,
where each clause is a disjunction of literals chosen independently and
uniformly. Our goal is to sample an approximately uniform solution of
(or equivalently, approximate the partition function of ).
Let be the density. The previous best algorithm runs in time
for any [Galanis,
Goldberg, Guo, and Yang, SIAM J. Comput.'21]. Our result significantly improves
both bounds by providing an almost-linear time sampler for any
.
The density captures the \emph{average degree} in the random
formula. In the worst-case model with bounded \emph{maximum degree}, current
best efficient sampler works up to degree bound [He, Wang, and Yin,
FOCS'22 and SODA'23], which is, for the first time, superseded by its
average-case counterpart due to our bound. Our result is the first
progress towards establishing the intuition that the solvability of the
average-case model (random -CNF formula with bounded average degree) is
better than the worst-case model (standard -CNF formula with bounded maximal
degree) in terms of sampling solutions.Comment: 51 pages, all proofs added, and bounds slightly improve