Coloring random graphs

We study the graph coloring problem over random graphs of finite average connectivity $c$. Given a number $q$ 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 $q$, we find the precise value of the critical average connectivity $c_q$. Moreover, we show that below $c_q$ there exist a clustering phase $c\in [c_d,c_q]$ 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

Polynomial iterative algorithms for coloring and analyzing random graphs

We study the graph coloring problem over random graphs of finite average connectivity $c$. Given a number $q$ 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 $q$, we find the precise value of the critical average connectivity $c_q$. Moreover, we show that below $c_q$ there exist a clustering phase $c\in [c_d,c_q]$ in which ground states spontaneously divide into an exponential number of clusters. Furthermore, we extended our considerations to the case of single instances showing consistent results. This lead us to propose a new algorithm able to color in polynomial time random graphs in the hard but colorable region, i.e when $c\in [c_d,c_q]$.Comment: 23 pages, 10 eps figure

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

Replica Cluster Variational Method: the Replica Symmetric solution for the 2D random bond Ising model

We present and solve the Replica Symmetric equations in the context of the Replica Cluster Variational Method for the 2D random bond Ising model (including the 2D Edwards-Anderson spin glass model). First we solve a linearized version of these equations to obtain the phase diagrams of the model on the square and triangular lattices. In both cases the spin-glass transition temperatures and the tricritical point estimations improve largely over the Bethe predictions. Moreover, we show that this phase diagram is consistent with the behavior of inference algorithms on single instances of the problem. Finally, we present a method to consistently find approximate solutions to the equations in the glassy phase. The method is applied to the triangular lattice down to T=0, also in the presence of an external field.Comment: 22 pages, 11 figure

Dynamics of systems with isotropic competing interactions in an external field: a Langevin approach

We study the Langevin dynamics of a ferromagnetic Ginzburg-Landau Hamiltonian with a competing long-range repulsive term in the presence of an external magnetic field. The model is analytically solved within the self consistent Hartree approximation for two different initial conditions: disordered or zero field cooled (ZFC), and fully magnetized or field cooled (FC). To test the predictions of the approximation we develop a suitable numerical scheme to ensure the isotropic nature of the interactions. Both the analytical approach and the numerical simulations of two-dimensional finite systems confirm a simple aging scenario at zero temperature and zero field. At zero temperature a critical field $h_c$ is found below which the initial conditions are relevant for the long time dynamics of the system. For $h < h_c$ a logarithmic growth of modulated domains is found in the numerical simulations but this behavior is not captured by the analytical approach which predicts a $t^1/2$ growth law at $T = 0$

Characterizing and Improving Generalized Belief Propagation Algorithms on the 2D Edwards-Anderson Model

We study the performance of different message passing algorithms in the two dimensional Edwards Anderson model. We show that the standard Belief Propagation (BP) algorithm converges only at high temperature to a paramagnetic solution. Then, we test a Generalized Belief Propagation (GBP) algorithm, derived from a Cluster Variational Method (CVM) at the plaquette level. We compare its performance with BP and with other algorithms derived under the same approximation: Double Loop (DL) and a two-ways message passing algorithm (HAK). The plaquette-CVM approximation improves BP in at least three ways: the quality of the paramagnetic solution at high temperatures, a better estimate (lower) for the critical temperature, and the fact that the GBP message passing algorithm converges also to non paramagnetic solutions. The lack of convergence of the standard GBP message passing algorithm at low temperatures seems to be related to the implementation details and not to the appearance of long range order. In fact, we prove that a gauge invariance of the constrained CVM free energy can be exploited to derive a new message passing algorithm which converges at even lower temperatures. In all its region of convergence this new algorithm is faster than HAK and DL by some orders of magnitude.Comment: 19 pages, 13 figure

Magnetic field chaos in the SK Model

We study the Sherrington--Kirkpatrick model, both above and below the De Almeida Thouless line, by using a modified version of the Parallel Tempering algorithm in which the system is allowed to move between different values of the magnetic field h. The behavior of the probability distribution of the overlap between two replicas at different values of the magnetic field h_0 and h_1 gives clear evidence for the presence of magnetic field chaos already for moderate system sizes, in contrast to the case of temperature chaos, which is not visible on system sizes that can currently be thermalized.Comment: Latex, 16 pages including 20 postscript figure

Coupled surface polaritons and the Casimir force

The Casimir force between metallic plates made of realistic materials is evaluated for distances in the nanometer range. A spectrum over real frequencies is introduced and shows narrow peaks due to surface resonances (plasmon polaritons or phonon polaritons) that are coupled across the vacuum gap. We demonstrate that the Casimir force originates from the attraction (repulsion) due to the corresponding symmetric (antisymmetric) eigenmodes, respectively. This picture is used to derive a simple analytical estimate of the Casimir force at short distances. We recover the result known for Drude metals without absorption and compute the correction for weakly absorbing materials.Comment: revised version submitted to Phys. Rev. A, 06 November 200

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