1,538 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
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
Random subcubes as a toy model for constraint satisfaction problems
We present an exactly solvable random-subcube model inspired by the structure
of hard constraint satisfaction and optimization problems. Our model reproduces
the structure of the solution space of the random k-satisfiability and
k-coloring problems, and undergoes the same phase transitions as these
problems. The comparison becomes quantitative in the large-k limit. Distance
properties, as well the x-satisfiability threshold, are studied. The model is
also generalized to define a continuous energy landscape useful for studying
several aspects of glassy dynamics.Comment: 21 pages, 4 figure
Computing a Knot Invariant as a Constraint Satisfaction Problem
We point out the connection between mathematical knot theory and spin
glass/search problem. In particular, we present a statistical mechanical
formulation of the problem of computing a knot invariant; p-colorability
problem, which provides an algorithm to find the solution. The method also
allows one to get some deeper insight into the structural complexity of knots,
which is expected to be related with the landscape structure of constraint
satisfaction problem.Comment: 6 pages, 3 figures, submitted to short note in Journal of Physical
Society of Japa
Classical-Quantum Mixing in the Random 2-Satisfiability Problem
Classical satisfiability (SAT) and quantum satisfiability (QSAT) are complete
problems for the complexity classes NP and QMA which are believed to be
intractable for classical and quantum computers, respectively. Statistical
ensembles of instances of these problems have been studied previously in an
attempt to elucidate their typical, as opposed to worst case, behavior. In this
paper we introduce a new statistical ensemble that interpolates between
classical and quantum. For the simplest 2-SAT/2-QSAT ensemble we find the exact
boundary that separates SAT and UNSAT instances. We do so by establishing
coincident lower and upper bounds, in the limit of large instances, on the
extent of the UNSAT and SAT regions, respectively.Comment: Updated reference
Simplest random K-satisfiability problem
We study a simple and exactly solvable model for the generation of random
satisfiability problems. These consist of random boolean constraints
which are to be satisfied simultaneously by logical variables. In
statistical-mechanics language, the considered model can be seen as a diluted
p-spin model at zero temperature. While such problems become extraordinarily
hard to solve by local search methods in a large region of the parameter space,
still at least one solution may be superimposed by construction. The
statistical properties of the model can be studied exactly by the replica
method and each single instance can be analyzed in polynomial time by a simple
global solution method. The geometrical/topological structures responsible for
dynamic and static phase transitions as well as for the onset of computational
complexity in local search method are thoroughly analyzed. Numerical analysis
on very large samples allows for a precise characterization of the critical
scaling behaviour.Comment: 14 pages, 5 figures, to appear in Phys. Rev. E (Feb 2001). v2: minor
errors and references correcte
Biased landscapes for random Constraint Satisfaction Problems
The typical complexity of Constraint Satisfaction Problems (CSPs) can be
investigated by means of random ensembles of instances. The latter exhibit many
threshold phenomena besides their satisfiability phase transition, in
particular a clustering or dynamic phase transition (related to the tree
reconstruction problem) at which their typical solutions shatter into
disconnected components. In this paper we study the evolution of this
phenomenon under a bias that breaks the uniformity among solutions of one CSP
instance, concentrating on the bicoloring of k-uniform random hypergraphs. We
show that for small k the clustering transition can be delayed in this way to
higher density of constraints, and that this strategy has a positive impact on
the performances of Simulated Annealing algorithms. We characterize the modest
gain that can be expected in the large k limit from the simple implementation
of the biasing idea studied here. This paper contains also a contribution of a
more methodological nature, made of a review and extension of the methods to
determine numerically the discontinuous dynamic transition threshold.Comment: 32 pages, 16 figure
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