35,490 research outputs found
Reweighted belief propagation and quiet planting for random K-SAT
We study the random K-satisfiability problem using a partition function where
each solution is reweighted according to the number of variables that satisfy
every clause. We apply belief propagation and the related cavity method to the
reweighted partition function. This allows us to obtain several new results on
the properties of random K-satisfiability problem. In particular the
reweighting allows to introduce a planted ensemble that generates instances
that are, in some region of parameters, equivalent to random instances. We are
hence able to generate at the same time a typical random SAT instance and one
of its solutions. We study the relation between clustering and belief
propagation fixed points and we give a direct evidence for the existence of
purely entropic (rather than energetic) barriers between clusters in some
region of parameters in the random K-satisfiability problem. We exhibit, in
some large planted instances, solutions with a non-trivial whitening core; such
solutions were known to exist but were so far never found on very large
instances. Finally, we discuss algorithmic hardness of such planted instances
and we determine a region of parameters in which planting leads to satisfiable
benchmarks that, up to our knowledge, are the hardest known.Comment: 23 pages, 4 figures, revised for readability, stability expression
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Quiet Planting in the Locked Constraint Satisfaction Problems
We study the planted ensemble of locked constraint satisfaction problems. We
describe the connection between the random and planted ensembles. The use of
the cavity method is combined with arguments from reconstruction on trees and
first and second moment considerations; in particular the connection with the
reconstruction on trees appears to be crucial. Our main result is the location
of the hard region in the planted ensemble. In a part of that hard region
instances have with high probability a single satisfying assignment.Comment: 21 pages, revised versio
Optimal Testing for Planted Satisfiability Problems
We study the problem of detecting planted solutions in a random
satisfiability formula. Adopting the formalism of hypothesis testing in
statistical analysis, we describe the minimax optimal rates of detection. Our
analysis relies on the study of the number of satisfying assignments, for which
we prove new results. We also address algorithmic issues, and give a
computationally efficient test with optimal statistical performance. This
result is compared to an average-case hypothesis on the hardness of refuting
satisfiability of random formulas
Spectral Detection on Sparse Hypergraphs
We consider the problem of the assignment of nodes into communities from a
set of hyperedges, where every hyperedge is a noisy observation of the
community assignment of the adjacent nodes. We focus in particular on the
sparse regime where the number of edges is of the same order as the number of
vertices. We propose a spectral method based on a generalization of the
non-backtracking Hashimoto matrix into hypergraphs. We analyze its performance
on a planted generative model and compare it with other spectral methods and
with Bayesian belief propagation (which was conjectured to be asymptotically
optimal for this model). We conclude that the proposed spectral method detects
communities whenever belief propagation does, while having the important
advantages to be simpler, entirely nonparametric, and to be able to learn the
rule according to which the hyperedges were generated without prior
information.Comment: 8 pages, 5 figure
From one solution of a 3-satisfiability formula to a solution cluster: Frozen variables and entropy
A solution to a 3-satisfiability (3-SAT) formula can be expanded into a
cluster, all other solutions of which are reachable from this one through a
sequence of single-spin flips. Some variables in the solution cluster are
frozen to the same spin values by one of two different mechanisms: frozen-core
formation and long-range frustrations. While frozen cores are identified by a
local whitening algorithm, long-range frustrations are very difficult to trace,
and they make an entropic belief-propagation (BP) algorithm fail to converge.
For BP to reach a fixed point the spin values of a tiny fraction of variables
(chosen according to the whitening algorithm) are externally fixed during the
iteration. From the calculated entropy values, we infer that, for a large
random 3-SAT formula with constraint density close to the satisfiability
threshold, the solutions obtained by the survey-propagation or the walksat
algorithm belong neither to the most dominating clusters of the formula nor to
the most abundant clusters. This work indicates that a single solution cluster
of a random 3-SAT formula may have further community structures.Comment: 13 pages, 6 figures. Final version as published in PR
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