45 research outputs found
On the cavity method for decimated random constraint satisfaction problems and the analysis of belief propagation guided decimation algorithms
We introduce a version of the cavity method for diluted mean-field spin
models that allows the computation of thermodynamic quantities similar to the
Franz-Parisi quenched potential in sparse random graph models. This method is
developed in the particular case of partially decimated random constraint
satisfaction problems. This allows to develop a theoretical understanding of a
class of algorithms for solving constraint satisfaction problems, in which
elementary degrees of freedom are sequentially assigned according to the
results of a message passing procedure (belief-propagation). We confront this
theoretical analysis to the results of extensive numerical simulations.Comment: 32 pages, 24 figure
Approaching the Rate-Distortion Limit with Spatial Coupling, Belief propagation and Decimation
We investigate an encoding scheme for lossy compression of a binary symmetric
source based on simple spatially coupled Low-Density Generator-Matrix codes.
The degree of the check nodes is regular and the one of code-bits is Poisson
distributed with an average depending on the compression rate. The performance
of a low complexity Belief Propagation Guided Decimation algorithm is
excellent. The algorithmic rate-distortion curve approaches the optimal curve
of the ensemble as the width of the coupling window grows. Moreover, as the
check degree grows both curves approach the ultimate Shannon rate-distortion
limit. The Belief Propagation Guided Decimation encoder is based on the
posterior measure of a binary symmetric test-channel. This measure can be
interpreted as a random Gibbs measure at a "temperature" directly related to
the "noise level of the test-channel". We investigate the links between the
algorithmic performance of the Belief Propagation Guided Decimation encoder and
the phase diagram of this Gibbs measure. The phase diagram is investigated
thanks to the cavity method of spin glass theory which predicts a number of
phase transition thresholds. In particular the dynamical and condensation
"phase transition temperatures" (equivalently test-channel noise thresholds)
are computed. We observe that: (i) the dynamical temperature of the spatially
coupled construction saturates towards the condensation temperature; (ii) for
large degrees the condensation temperature approaches the temperature (i.e.
noise level) related to the information theoretic Shannon test-channel noise
parameter of rate-distortion theory. This provides heuristic insight into the
excellent performance of the Belief Propagation Guided Decimation algorithm.
The paper contains an introduction to the cavity method
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
Phase Transitions and Computational Difficulty in Random Constraint Satisfaction Problems
We review the understanding of the random constraint satisfaction problems,
focusing on the q-coloring of large random graphs, that has been achieved using
the cavity method of the physicists. We also discuss the properties of the
phase diagram in temperature, the connections with the glass transition
phenomenology in physics, and the related algorithmic issues.Comment: 10 pages, Proceedings of the International Workshop on
Statistical-Mechanical Informatics 2007, Kyoto (Japan) September 16-19, 200
The backtracking survey propagation algorithm for solving random K-SAT problems
Discrete combinatorial optimization has a central role in many scientific
disciplines, however, for hard problems we lack linear time algorithms that
would allow us to solve very large instances. Moreover, it is still unclear
what are the key features that make a discrete combinatorial optimization
problem hard to solve. Here we study random K-satisfiability problems with
, which are known to be very hard close to the SAT-UNSAT threshold,
where problems stop having solutions. We show that the backtracking survey
propagation algorithm, in a time practically linear in the problem size, is
able to find solutions very close to the threshold, in a region unreachable by
any other algorithm. All solutions found have no frozen variables, thus
supporting the conjecture that only unfrozen solutions can be found in linear
time, and that a problem becomes impossible to solve in linear time when all
solutions contain frozen variables.Comment: 11 pages, 10 figures. v2: data largely improved and manuscript
rewritte
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
correcte
The decimation process in random k-SAT
Let F be a uniformly distributed random k-SAT formula with n variables and m
clauses. Non-rigorous statistical mechanics ideas have inspired a message
passing algorithm called Belief Propagation Guided Decimation for finding
satisfying assignments of F. This algorithm can be viewed as an attempt at
implementing a certain thought experiment that we call the Decimation Process.
In this paper we identify a variety of phase transitions in the decimation
process and link these phase transitions to the performance of the algorithm
Comparing Beliefs, Surveys and Random Walks
Survey propagation is a powerful technique from statistical physics that has
been applied to solve the 3-SAT problem both in principle and in practice. We
give, using only probability arguments, a common derivation of survey
propagation, belief propagation and several interesting hybrid methods. We then
present numerical experiments which use WSAT (a widely used random-walk based
SAT solver) to quantify the complexity of the 3-SAT formulae as a function of
their parameters, both as randomly generated and after simplification, guided
by survey propagation. Some properties of WSAT which have not previously been
reported make it an ideal tool for this purpose -- its mean cost is
proportional to the number of variables in the formula (at a fixed ratio of
clauses to variables) in the easy-SAT regime and slightly beyond, and its
behavior in the hard-SAT regime appears to reflect the underlying structure of
the solution space that has been predicted by replica symmetry-breaking
arguments. An analysis of the tradeoffs between the various methods of search
for satisfying assignments shows WSAT to be far more powerful that has been
appreciated, and suggests some interesting new directions for practical
algorithm development.Comment: 8 pages, 5 figure