366 research outputs found
Solving satisfiability problems by fluctuations: The dynamics of stochastic local search algorithms
Stochastic local search algorithms are frequently used to numerically solve
hard combinatorial optimization or decision problems. We give numerical and
approximate analytical descriptions of the dynamics of such algorithms applied
to random satisfiability problems. We find two different dynamical regimes,
depending on the number of constraints per variable: For low constraintness,
the problems are solved efficiently, i.e. in linear time. For higher
constraintness, the solution times become exponential. We observe that the
dynamical behavior is characterized by a fast equilibration and fluctuations
around this equilibrium. If the algorithm runs long enough, an exponentially
rare fluctuation towards a solution appears.Comment: 21 pages, 18 figures, revised version, to app. in PRE (2003
Boosting search by rare events
Randomized search algorithms for hard combinatorial problems exhibit a large
variability of performances. We study the different types of rare events which
occur in such out-of-equilibrium stochastic processes and we show how they
cooperate in determining the final distribution of running times. As a
byproduct of our analysis we show how search algorithms are optimized by random
restarts.Comment: 4 pages, 3 eps figures. References update
Computational complexity arising from degree correlations in networks
We apply a Bethe-Peierls approach to statistical-mechanics models defined on
random networks of arbitrary degree distribution and arbitrary correlations
between the degrees of neighboring vertices. Using the NP-hard optimization
problem of finding minimal vertex covers on these graphs, we show that such
correlations may lead to a qualitatively different solution structure as
compared to uncorrelated networks. This results in a higher complexity of the
network in a computational sense: Simple heuristic algorithms fail to find a
minimal vertex cover in the highly correlated case, whereas uncorrelated
networks seem to be simple from the point of view of combinatorial
optimization.Comment: 4 pages, 1 figure, accepted in Phys. Rev.
Multifractal analysis of perceptron learning with errors
Random input patterns induce a partition of the coupling space of a
perceptron into cells labeled by their output sequences. Learning some data
with a maximal error rate leads to clusters of neighboring cells. By analyzing
the internal structure of these clusters with the formalism of multifractals,
we can handle different storage and generalization tasks for lazy students and
absent-minded teachers within one unified approach. The results also allow some
conclusions on the spatial distribution of cells.Comment: 11 pages, RevTex, 3 eps figures, version to be published in Phys.
Rev. E 01Jan9
Random Graph Coloring - a Statistical Physics Approach
The problem of vertex coloring in random graphs is studied using methods of
statistical physics and probability. Our analytical results are compared to
those obtained by exact enumeration and Monte-Carlo simulations. We critically
discuss the merits and shortcomings of the various methods, and interpret the
results obtained. We present an exact analytical expression for the 2-coloring
problem as well as general replica symmetric approximated solutions for the
thermodynamics of the graph coloring problem with p colors and K-body edges.Comment: 17 pages, 9 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
Exponentially hard problems are sometimes polynomial, a large deviation analysis of search algorithms for the random Satisfiability problem, and its application to stop-and-restart resolutions
A large deviation analysis of the solving complexity of random
3-Satisfiability instances slightly below threshold is presented. While finding
a solution for such instances demands an exponential effort with high
probability, we show that an exponentially small fraction of resolutions
require a computation scaling linearly in the size of the instance only. This
exponentially small probability of easy resolutions is analytically calculated,
and the corresponding exponent shown to be smaller (in absolute value) than the
growth exponent of the typical resolution time. Our study therefore gives some
theoretical basis to heuristic stop-and-restart solving procedures, and
suggests a natural cut-off (the size of the instance) for the restart.Comment: Revtex file, 4 figure
Stability of the replica-symmetric saddle-point in general mean-field spin-glass models
Within the replica approach to mean-field spin-glasses the transition from
ergodic high-temperature behaviour to the glassy low-temperature phase is
marked by the instability of the replica-symmetric saddle-point. For general
spin-glass models with non-Gaussian field distributions the corresponding
Hessian is a matrix with the number of replicas tending to
zero eventually. We block-diagonalize this Hessian matrix using representation
theory of the permutation group and identify the blocks related to the
spin-glass susceptibility. Performing the limit within these blocks we
derive expressions for the de~Almeida-Thouless line of general spin-glass
models. Specifying these expressions to the cases of the
Sherrington-Kirkpatrick, Viana-Bray, and the L\'evy spin glass respectively we
obtain results in agreement with previous findings using the cavity approach
Coloring random graphs
We study the graph coloring problem over random graphs of finite average
connectivity . Given a number of available colors, we find that graphs
with low connectivity admit almost always a proper coloring whereas graphs with
high connectivity are uncolorable. Depending on , we find the precise value
of the critical average connectivity . Moreover, we show that below
there exist a clustering phase in which ground states
spontaneously divide into an exponential number of clusters and where the
proliferation of metastable states is responsible for the onset of complexity
in local search algorithms.Comment: 4 pages, 1 figure, version to app. in PR
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