112 research outputs found
Systematic perturbation approach for a dynamical scaling law in a kinetically constrained spin model
The dynamical behaviours of a kinetically constrained spin model
(Fredrickson-Andersen model) on a Bethe lattice are investigated by a
perturbation analysis that provides exact final states above the nonergodic
transition point. It is observed that the time-dependent solutions of the
derived dynamical systems obtained by the perturbation analysis become
systematically closer to the results obtained by Monte Carlo simulations as the
order of a perturbation series is increased. This systematic perturbation
analysis also clarifies the existence of a dynamical scaling law, which
provides a implication for a universal relation between a size scale and a time
scale near the nonergodic transition.Comment: 17 pages, 7 figures, v2; results have been refined, v3; A figure has
been modified, v4; results have been more refine
Relationship between clustering and algorithmic phase transitions in the random k-XORSAT model and its NP-complete extensions
We study the performances of stochastic heuristic search algorithms on
Uniquely Extendible Constraint Satisfaction Problems with random inputs. We
show that, for any heuristic preserving the Poissonian nature of the underlying
instance, the (heuristic-dependent) largest ratio of constraints per
variables for which a search algorithm is likely to find solutions is smaller
than the critical ratio above which solutions are clustered and
highly correlated. In addition we show that the clustering ratio can be reached
when the number k of variables per constraints goes to infinity by the
so-called Generalized Unit Clause heuristic.Comment: 15 pages, 4 figures, Proceedings of the International Workshop on
Statistical-Mechanical Informatics, September 16-19, 2007, Kyoto, Japan; some
imprecisions in the previous version have been correcte
Message passing for vertex covers
Constructing a minimal vertex cover of a graph can be seen as a prototype for
a combinatorial optimization problem under hard constraints. In this paper, we
develop and analyze message passing techniques, namely warning and survey
propagation, which serve as efficient heuristic algorithms for solving these
computational hard problems. We show also, how previously obtained results on
the typical-case behavior of vertex covers of random graphs can be recovered
starting from the message passing equations, and how they can be extended.Comment: 25 pages, 9 figures - version accepted for publication in PR
Geometrical organization of solutions to random linear Boolean equations
The random XORSAT problem deals with large random linear systems of Boolean
variables. The difficulty of such problems is controlled by the ratio of number
of equations to number of variables. It is known that in some range of values
of this parameter, the space of solutions breaks into many disconnected
clusters. Here we study precisely the corresponding geometrical organization.
In particular, the distribution of distances between these clusters is computed
by the cavity method. This allows to study the `x-satisfiability' threshold,
the critical density of equations where there exist two solutions at a given
distance.Comment: 20 page
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
Reconstruction of Random Colourings
Reconstruction problems have been studied in a number of contexts including
biology, information theory and and statistical physics. We consider the
reconstruction problem for random -colourings on the -ary tree for
large . Bhatnagar et. al. showed non-reconstruction when and reconstruction when . We tighten this result and show non-reconstruction when and reconstruction when .Comment: Added references, updated notatio
A hard-sphere model on generalized Bethe lattices: Statics
We analyze the phase diagram of a model of hard spheres of chemical radius
one, which is defined over a generalized Bethe lattice containing short loops.
We find a liquid, two different crystalline, a glassy and an unusual
crystalline glassy phase. Special attention is also paid to the close-packing
limit in the glassy phase. All analytical results are cross-checked by
numerical Monte-Carlo simulations.Comment: 24 pages, revised versio
Spin models on random graphs with controlled topologies beyond degree constraints
We study Ising spin models on finitely connected random interaction graphs
which are drawn from an ensemble in which not only the degree distribution
can be chosen arbitrarily, but which allows for further fine-tuning of
the topology via preferential attachment of edges on the basis of an arbitrary
function Q(k,k') of the degrees of the vertices involved. We solve these models
using finite connectivity equilibrium replica theory, within the replica
symmetric ansatz. In our ensemble of graphs, phase diagrams of the spin system
are found to depend no longer only on the chosen degree distribution, but also
on the choice made for Q(k,k'). The increased ability to control interaction
topology in solvable models beyond prescribing only the degree distribution of
the interaction graph enables a more accurate modeling of real-world
interacting particle systems by spin systems on suitably defined random graphs.Comment: 21 pages, 4 figures, submitted to J Phys
Focused Local Search for Random 3-Satisfiability
A local search algorithm solving an NP-complete optimisation problem can be
viewed as a stochastic process moving in an 'energy landscape' towards
eventually finding an optimal solution. For the random 3-satisfiability
problem, the heuristic of focusing the local moves on the presently
unsatisfiedclauses is known to be very effective: the time to solution has been
observed to grow only linearly in the number of variables, for a given
clauses-to-variables ratio sufficiently far below the critical
satisfiability threshold . We present numerical results
on the behaviour of three focused local search algorithms for this problem,
considering in particular the characteristics of a focused variant of the
simple Metropolis dynamics. We estimate the optimal value for the
``temperature'' parameter for this algorithm, such that its linear-time
regime extends as close to as possible. Similar parameter
optimisation is performed also for the well-known WalkSAT algorithm and for the
less studied, but very well performing Focused Record-to-Record Travel method.
We observe that with an appropriate choice of parameters, the linear time
regime for each of these algorithms seems to extend well into ratios -- much further than has so far been generally assumed. We discuss the
statistics of solution times for the algorithms, relate their performance to
the process of ``whitening'', and present some conjectures on the shape of
their computational phase diagrams.Comment: 20 pages, lots of figure
Exhaustive enumeration unveils clustering and freezing in random 3-SAT
We study geometrical properties of the complete set of solutions of the
random 3-satisfiability problem. We show that even for moderate system sizes
the number of clusters corresponds surprisingly well with the theoretic
asymptotic prediction. We locate the freezing transition in the space of
solutions which has been conjectured to be relevant in explaining the onset of
computational hardness in random constraint satisfaction problems.Comment: 4 pages, 3 figure
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