4,662 research outputs found
Phase Transitions of the Typical Algorithmic Complexity of the Random Satisfiability Problem Studied with Linear Programming
Here we study the NP-complete -SAT problem. Although the worst-case
complexity of NP-complete problems is conjectured to be exponential, there
exist parametrized random ensembles of problems where solutions can typically
be found in polynomial time for suitable ranges of the parameter. In fact,
random -SAT, with as control parameter, can be solved quickly
for small enough values of . It shows a phase transition between a
satisfiable phase and an unsatisfiable phase. For branch and bound algorithms,
which operate in the space of feasible Boolean configurations, the empirically
hardest problems are located only close to this phase transition. Here we study
-SAT () and the related optimization problem MAX-SAT by a linear
programming approach, which is widely used for practical problems and allows
for polynomial run time. In contrast to branch and bound it operates outside
the space of feasible configurations. On the other hand, finding a solution
within polynomial time is not guaranteed. We investigated several variants like
including artificial objective functions, so called cutting-plane approaches,
and a mapping to the NP-complete vertex-cover problem. We observed several
easy-hard transitions, from where the problems are typically solvable (in
polynomial time) using the given algorithms, respectively, to where they are
not solvable in polynomial time. For the related vertex-cover problem on random
graphs these easy-hard transitions can be identified with structural properties
of the graphs, like percolation transitions. For the present random -SAT
problem we have investigated numerous structural properties also exhibiting
clear transitions, but they appear not be correlated to the here observed
easy-hard transitions. This renders the behaviour of random -SAT more
complex than, e.g., the vertex-cover problem.Comment: 11 pages, 5 figure
Exactly solvable models of adaptive networks
A satisfiability (SAT-UNSAT) transition takes place for many optimization
problems when the number of constraints, graphically represented by links
between variables nodes, is brought above some threshold. If the network of
constraints is allowed to adapt by redistributing its links, the SAT-UNSAT
transition may be delayed and preceded by an intermediate phase where the
structure self-organizes to satisfy the constraints. We present an analytic
approach, based on the recently introduced cavity method for large deviations,
which exactly describes the two phase transitions delimiting this adaptive
intermediate phase. We give explicit results for random bond models subject to
the connectivity or rigidity percolation transitions, and compare them with
numerical simulations.Comment: 4 pages, 4 figure
Cluster expansions in dilute systems: applications to satisfiability problems and spin glasses
We develop a systematic cluster expansion for dilute systems in the highly
dilute phase. We first apply it to the calculation of the entropy of the
K-satisfiability problem in the satisfiable phase. We derive a series expansion
in the control parameter, the average connectivity, that is identical to the
one obtained by using the replica approach with a replica symmetric ({\sc rs})
{\it Ansatz}, when the order parameter is calculated via a perturbative
expansion in the control parameter. As a second application we compute the
free-energy of the Viana-Bray model in the paramagnetic phase. The cluster
expansion allows one to compute finite-size corrections in a simple manner and
these are particularly important in optimization problems. Importantly enough,
these calculations prove the exactness of the {\sc rs} {\it Ansatz} below the
percolation threshold and might require its revision between this and the
easy-to-hard transition.Comment: 21 pages, 7 figs, to appear in Phys. Rev.
Percolation of satisfiability in finite dimensions
The satisfiability and optimization of finite-dimensional Boolean formulas
are studied using percolation theory, rare region arguments, and boundary
effects. In contrast with mean-field results, there is no satisfiability
transition, though there is a logical connectivity transition. In part of the
disconnected phase, rare regions lead to a divergent running time for
optimization algorithms. The thermodynamic ground state for the NP-hard
two-dimensional maximum-satisfiability problem is typically unique. These
results have implications for the computational study of disordered materials.Comment: 4 pages, 4 fig
Scale-Free Random SAT Instances
We focus on the random generation of SAT instances that have properties
similar to real-world instances. It is known that many industrial instances,
even with a great number of variables, can be solved by a clever solver in a
reasonable amount of time. This is not possible, in general, with classical
randomly generated instances. We provide a different generation model of SAT
instances, called \emph{scale-free random SAT instances}. It is based on the
use of a non-uniform probability distribution to select
variable , where is a parameter of the model. This results into
formulas where the number of occurrences of variables follows a power-law
distribution where . This property
has been observed in most real-world SAT instances. For , our model
extends classical random SAT instances.
We prove the existence of a SAT-UNSAT phase transition phenomenon for
scale-free random 2-SAT instances with when the clause/variable
ratio is . We also prove that scale-free
random k-SAT instances are unsatisfiable with high probability when the number
of clauses exceeds . %This implies that the SAT/UNSAT
phase transition phenomena vanishes when , and formulas are
unsatisfiable due to a small core of clauses. The proof of this result suggests
that, when , the unsatisfiability of most formulas may be due to
small cores of clauses. Finally, we show how this model will allow us to
generate random instances similar to industrial instances, of interest for
testing purposes
Statistical mechanics of the vertex-cover problem
We review recent progress in the study of the vertex-cover problem (VC). VC
belongs to the class of NP-complete graph theoretical problems, which plays a
central role in theoretical computer science. On ensembles of random graphs, VC
exhibits an coverable-uncoverable phase transition. Very close to this
transition, depending on the solution algorithm, easy-hard transitions in the
typical running time of the algorithms occur.
We explain a statistical mechanics approach, which works by mapping VC to a
hard-core lattice gas, and then applying techniques like the replica trick or
the cavity approach. Using these methods, the phase diagram of VC could be
obtained exactly for connectivities , where VC is replica symmetric.
Recently, this result could be confirmed using traditional mathematical
techniques. For , the solution of VC exhibits full replica symmetry
breaking.
The statistical mechanics approach can also be used to study analytically the
typical running time of simple complete and incomplete algorithms for VC.
Finally, we describe recent results for VC when studied on other ensembles of
finite- and infinite-dimensional graphs.Comment: review article, 26 pages, 9 figures, to appear in J. Phys. A: Math.
Ge
Percolation on fitness landscapes: effects of correlation, phenotype, and incompatibilities
We study how correlations in the random fitness assignment may affect the
structure of fitness landscapes. We consider three classes of fitness models.
The first is a continuous phenotype space in which individuals are
characterized by a large number of continuously varying traits such as size,
weight, color, or concentrations of gene products which directly affect
fitness. The second is a simple model that explicitly describes
genotype-to-phenotype and phenotype-to-fitness maps allowing for neutrality at
both phenotype and fitness levels and resulting in a fitness landscape with
tunable correlation length. The third is a class of models in which particular
combinations of alleles or values of phenotypic characters are "incompatible"
in the sense that the resulting genotypes or phenotypes have reduced (or zero)
fitness. This class of models can be viewed as a generalization of the
canonical Bateson-Dobzhansky-Muller model of speciation. We also demonstrate
that the discrete NK model shares some signature properties of models with high
correlations. Throughout the paper, our focus is on the percolation threshold,
on the number, size and structure of connected clusters, and on the number of
viable genotypes.Comment: 31 pages, 4 figures, 1 tabl
Absence of magnetic long range order in YCrSbO: bond-disorder induced magnetic frustration in a ferromagnetic pyrochlore
The consequences of nonmagnetic-ion dilution for the pyrochlore family
Y()O ( = magnetic ion, = nonmagnetic
ion) have been investigated. As a first step, we experimentally examine the
magnetic properties of YCrSbO ( = 0.5), in which the magnetic
sites (Cr) are percolative. Although the effective Cr-Cr spin exchange
is ferromagnetic, as evidenced by a positive Curie-Weiss temperature,
= 20.1(6) K, our high-resolution neutron powder
diffraction measurements detect no sign of magnetic long range order down to 2
K. In order to understand our observations, we performed numerical simulations
to study the bond-disorder introduced by the ionic size mismatch between
and . Based on these simulations, bond-disorder ( 0.23)
percolates well ahead of site-disorder ( 0.61). This model
successfully reproduces the critical region (0.2 < < 0.25) for the N\'eel
to spin glass phase transition in Zn(CrGa)O, where
the Cr/Ga-sublattice forms the same corner-sharing tetrahedral network as the
-sublattice in Y()O, and the rapid drop in
magnetically ordered moment in the N\'eel phase [Lee , Phys. Rev. B
77, 014405 (2008)]. Our study stresses the nonnegligible role of bond-disorder
on magnetic frustration, even in ferromagnets
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