478 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
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.
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
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
A Causal Algebra for Liouville Exponentials
A causal Poisson bracket algebra for Liouville exponentials on a cylinder is
derived using an exchange algebra for free fields describing the in and out
asymptotics. The causal algebra involves an even number of space-time points
with a minimum of four. A quantum realisation of the algebra is obtained which
preserves causality and the local form of non-equal time brackets.Comment: 10 page
Entropy and typical properties of Nash equilibria in two-player games
We use techniques from the statistical mechanics of disordered systems to
analyse the properties of Nash equilibria of bimatrix games with large random
payoff matrices. By means of an annealed bound, we calculate their number and
analyse the properties of typical Nash equilibria, which are exponentially
dominant in number. We find that a randomly chosen equilibrium realizes almost
always equal payoffs to either player. This value and the fraction of
strategies played at an equilibrium point are calculated as a function of the
correlation between the two payoff matrices. The picture is complemented by the
calculation of the properties of Nash equilibria in pure strategies.Comment: 6 pages, was "Self averaging of Nash equilibria in two player games",
main section rewritten, some new results, for additional information see
http://itp.nat.uni-magdeburg.de/~jberg/games.htm
Glassy behavior induced by geometrical frustration in a hard-core lattice gas model
We introduce a hard-core lattice-gas model on generalized Bethe lattices and
investigate analytically and numerically its compaction behavior. If
compactified slowly, the system undergoes a first-order crystallization
transition. If compactified much faster, the system stays in a meta-stable
liquid state and undergoes a glass transition under further compaction. We show
that this behavior is induced by geometrical frustration which appears due to
the existence of short loops in the generalized Bethe lattices. We also compare
our results to numerical simulations of a three-dimensional analog of the
model.Comment: 7 pages, 4 figures, revised versio
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
Inference algorithms for gene networks: a statistical mechanics analysis
The inference of gene regulatory networks from high throughput gene
expression data is one of the major challenges in systems biology. This paper
aims at analysing and comparing two different algorithmic approaches. The first
approach uses pairwise correlations between regulated and regulating genes; the
second one uses message-passing techniques for inferring activating and
inhibiting regulatory interactions. The performance of these two algorithms can
be analysed theoretically on well-defined test sets, using tools from the
statistical physics of disordered systems like the replica method. We find that
the second algorithm outperforms the first one since it takes into account
collective effects of multiple regulators
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