11 research outputs found
Relaxation and Metastability in the RandomWalkSAT search procedure
An analysis of the average properties of a local search resolution procedure
for the satisfaction of random Boolean constraints is presented. Depending on
the ratio alpha of constraints per variable, resolution takes a time T_res
growing linearly (T_res \sim tau(alpha) N, alpha < alpha_d) or exponentially
(T_res \sim exp(N zeta(alpha)), alpha > alpha_d) with the size N of the
instance. The relaxation time tau(alpha) in the linear phase is calculated
through a systematic expansion scheme based on a quantum formulation of the
evolution operator. For alpha > alpha_d, the system is trapped in some
metastable state, and resolution occurs from escape from this state through
crossing of a large barrier. An annealed calculation of the height zeta(alpha)
of this barrier is proposed. The polynomial/exponentiel cross-over alpha_d is
not related to the onset of clustering among solutions.Comment: 23 pages, 11 figures. A mistake in sec. IV.B has been correcte
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
The dynamics of proving uncolourability of large random graphs I. Symmetric Colouring Heuristic
We study the dynamics of a backtracking procedure capable of proving
uncolourability of graphs, and calculate its average running time T for sparse
random graphs, as a function of the average degree c and the number of vertices
N. The analysis is carried out by mapping the history of the search process
onto an out-of-equilibrium (multi-dimensional) surface growth problem. The
growth exponent of the average running time is quantitatively predicted, in
agreement with simulations.Comment: 5 figure
Approximation schemes for the dynamics of diluted spin models: the Ising ferromagnet on a Bethe lattice
We discuss analytical approximation schemes for the dynamics of diluted spin
models. The original dynamics of the complete set of degrees of freedom is
replaced by a hierarchy of equations including an increasing number of global
observables, which can be closed approximately at different levels of the
hierarchy. We illustrate this method on the simple example of the Ising
ferromagnet on a Bethe lattice, investigating the first three possible
closures, which are all exact in the long time limit, and which yield more and
more accurate predictions for the finite-time behavior. We also investigate the
critical region around the phase transition, and the behavior of two-time
correlation functions. We finally underline the close relationship between this
approach and the dynamical replica theory under the assumption of replica
symmetry.Comment: 21 pages, 5 figure
Instability of one-step replica-symmetry-broken phase in satisfiability problems
We reconsider the one-step replica-symmetry-breaking (1RSB) solutions of two
random combinatorial problems: k-XORSAT and k-SAT. We present a general method
for establishing the stability of these solutions with respect to further steps
of replica-symmetry breaking. Our approach extends the ideas of [A.Montanari
and F. Ricci-Tersenghi, Eur.Phys.J. B 33, 339 (2003)] to more general
combinatorial problems.
It turns out that 1RSB is always unstable at sufficiently small clauses
density alpha or high energy. In particular, the recent 1RSB solution to 3-SAT
is unstable at zero energy for alpha< alpha_m, with alpha_m\approx 4.153. On
the other hand, the SAT-UNSAT phase transition seems to be correctly described
within 1RSB.Comment: 26 pages, 7 eps figure
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
Random subcubes as a toy model for constraint satisfaction problems
We present an exactly solvable random-subcube model inspired by the structure
of hard constraint satisfaction and optimization problems. Our model reproduces
the structure of the solution space of the random k-satisfiability and
k-coloring problems, and undergoes the same phase transitions as these
problems. The comparison becomes quantitative in the large-k limit. Distance
properties, as well the x-satisfiability threshold, are studied. The model is
also generalized to define a continuous energy landscape useful for studying
several aspects of glassy dynamics.Comment: 21 pages, 4 figure
Aging dynamics of heterogeneous spin models
We investigate numerically the dynamics of three different spin models in the
aging regime. Each of these models is meant to be representative of a distinct
class of aging behavior: coarsening systems, discontinuous spin glasses, and
continuous spin glasses. In order to study dynamic heterogeneities induced by
quenched disorder, we consider single-spin observables for a given disorder
realization. In some simple cases we are able to provide analytical predictions
for single-spin response and correlation functions.
The results strongly depend upon the model considered. It turns out that, by
comparing the slow evolution of a few different degrees of freedom, one can
distinguish between different dynamic classes. As a conclusion we present the
general properties which can be induced from our results, and discuss their
relation with thermometric arguments.Comment: 39 pages, 36 figure
On large deviation properties of Erdos-Renyi random graphs
We show that large deviation properties of Erd\"os-R\'enyi random graphs can
be derived from the free energy of the -state Potts model of statistical
mechanics. More precisely the Legendre transform of the Potts free energy with
respect to is related to the component generating function of the graph
ensemble. This generalizes the well-known mapping between typical properties of
random graphs and the limit of the Potts free energy. For
exponentially rare graphs we explicitly calculate the number of components, the
size of the giant component, the degree distributions inside and outside the
giant component, and the distribution of small component sizes. We also perform
numerical simulations which are in very good agreement with our analytical
work. Finally we demonstrate how the same results can be derived by studying
the evolution of random graphs under the insertion of new vertices and edges,
without recourse to the thermodynamics of the Potts model.Comment: 38 pages, 9 figures, Latex2e, corrected and extended version
including numerical simulation result
On the freezing of variables in random constraint satisfaction problems
The set of solutions of random constraint satisfaction problems (zero energy
groundstates of mean-field diluted spin glasses) undergoes several structural
phase transitions as the amount of constraints is increased. This set first
breaks down into a large number of well separated clusters. At the freezing
transition, which is in general distinct from the clustering one, some
variables (spins) take the same value in all solutions of a given cluster. In
this paper we study the critical behavior around the freezing transition, which
appears in the unfrozen phase as the divergence of the sizes of the
rearrangements induced in response to the modification of a variable. The
formalism is developed on generic constraint satisfaction problems and applied
in particular to the random satisfiability of boolean formulas and to the
coloring of random graphs. The computation is first performed in random tree
ensembles, for which we underline a connection with percolation models and with
the reconstruction problem of information theory. The validity of these results
for the original random ensembles is then discussed in the framework of the
cavity method.Comment: 32 pages, 7 figure