1,549 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
Dynamics of heuristic optimization algorithms on random graphs
In this paper, the dynamics of heuristic algorithms for constructing small
vertex covers (or independent sets) of finite-connectivity random graphs is
analysed. In every algorithmic step, a vertex is chosen with respect to its
vertex degree. This vertex, and some environment of it, is covered and removed
from the graph. This graph reduction process can be described as a Markovian
dynamics in the space of random graphs of arbitrary degree distribution. We
discuss some solvable cases, including algorithms already analysed using
different techniques, and develop approximation schemes for more complicated
cases. The approximations are corroborated by numerical simulations.Comment: 19 pages, 3 figures, version to app. in EPJ
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