5,706 research outputs found
Random k-SAT and the Power of Two Choices
We study an Achlioptas-process version of the random k-SAT process: a bounded
number of k-clauses are drawn uniformly at random at each step, and exactly one
added to the growing formula according to a particular rule. We prove the
existence of a rule that shifts the satisfiability threshold. This extends a
well-studied area of probabilistic combinatorics (Achlioptas processes) to
random CSP's. In particular, while a rule to delay the 2-SAT threshold was
known previously, this is the first proof of a rule to shift the threshold of
k-SAT for k >= 3.
We then propose a gap decision problem based upon this semi-random model. The
aim of the problem is to investigate the hardness of the random k-SAT decision
problem, as opposed to the problem of finding an assignment or certificate of
unsatisfiability. Finally, we discuss connections to the study of Achlioptas
random graph processes.Comment: 13 page
An exactly solvable random satisfiability problem
We introduce a new model for the generation of random satisfiability
problems. It is an extension of the hyper-SAT model of Ricci-Tersenghi, Weigt
and Zecchina, which is a variant of the famous K-SAT model: it is extended to
q-state variables and relates to a different choice of the statistical
ensemble. The model has an exactly solvable statistic: the critical exponents
and scaling functions of the SAT/UNSAT transition are calculable at zero
temperature, with no need of replicas, also with exact finite-size corrections.
We also introduce an exact duality of the model, and show an analogy of
thermodynamic properties with the Random Energy Model of disordered spin
systems theory. Relations with Error-Correcting Codes are also discussed.Comment: 31 pages, 1 figur
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
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
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
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