internal diffusion limited aggregation parallel algorithms and complexity the computational complexity of internal diffusion limited aggregation dla is examined from both a theoretical and a practical point of view we show that for two or more dimensions the problem of predicting the cluster from a given set of paths is complete for the complexity class cc the subset of p characterized by circuits composed of comparator gates cc completeness is believed to imply that in the worst case growing a cluster of size n requires polynomial time in n even on a parallel computer a parallel relaxation algorithm is presented that uses the fact that clusters are nearly spherical to guess the cluster from a given set of paths and then corrects defects in the guessed cluster through a nonlocal annihilation process the parallel running time of the relaxation algorithm for two dimensional internal dla is studied by simulating it on a serial computer the numerical results are compatible with a running time that is either polylogarithmic in n or a small power of n thus the computational resources needed to grow large clusters are significantly less on average than the worst case analysis would suggest for a parallel machine with k processors we show that random clusters in d dimensions can be generated in mathcal o n k logk n 2 d steps this is a significant speedup over explicit sequential simulation which takes mathcal o n 1 2 d time on average finally we show that in one dimension internal dla can be predicted in mathcal o logn parallel time and so is in the complexity class nc

bounding the bias of contrastive divergence learning optimization based on k step contrastive divergence cd has become a common way to train restricted boltzmann machines rbms the k step cd is a biased estimator of the log likelihood gradient relying on gibbs sampling we derive a new upper bound for this bias its magnitude depends on k the number of variables in the rbm and the maximum change in energy that can be produced by changing a single variable the last reflects the dependence on the absolute values of the rbm parameters the magnitude of the bias is also affected by the distance in variation between the modeled distribution and the starting distribution of the gibbs chain

on the computational complexity of ising spin glass models in a spin glass with ising spins the problems of computing the magnetic partition function and finding a ground state are studied in a finite two dimensional lattice these problems can be solved by algorithms that require a number of steps bounded by a polynomial function of the size of the lattice in contrast to this fact the same problems are shown to belong to the class of np hard problems both in the two dimensional case within a magnetic field and in the three dimensional case np hardness of a problem suggests that it is very unlikely that a polynomial algorithm could exist to solve it

asymptotics of work distributions in nonequilibrium systems the asymptotic behavior of the work distribution in driven nonequilibrium systems is determined using the method of optimal fluctuations for systems described by langevin dynamics the corresponding euler lagrange equation together with the appropriate boundary conditions and an equation for the leading pre exponential factor are derived the method is applied to three representative examples and the results are used to improve the accuracy of free energy estimates based on the application of the jarzynski equation

proposition of a new approach for the substructure method abstract in this work a new projection basis for the substructure method is presented this basis uses new correction solutions reducing the numerical of co ordinates used in the dynamic analysis of a complex structure for each connexion boundary a boundary structure whose characteristics are defined using a physical criterion is associated the correction solutions proposed here use the normal modes of the boundary structure on this basis the size of problem and the compute time are reduced two structures are studied a rectangular plate and the planetarium of the science city of tunisia the results show that the proposed method gives a good estimation of frequencies in comparison with the full finite element method results

a general procedure for improving substructures representation in dynamic synthesis in the general substructure synthesis method for the dynamic analysis of complex flexible structures developed earlier by the authors the motion of each substructure is represented by a given number of substructure admissible functions and the otherwise disjoint substructures are connected together to form a whole structure by imposing approximate geometric compatibility conditions by means of weighted residuals in this paper a general procedure for improving the admissible functions representing each substructure in the synthesis is developed the procedure is an iterative one and it leads to an improved eigensolution for the intermediate structure i e a fictitious structure defined by the approximate compatibility conditions where the improved eigensolution is obtained while always using the same number of degrees of freedom to represent a given substructure in any iteration the computations associated with each substructure are independent of those for all other substructures and they can be performed in parallel by increasing the number of iterations the eigensolution for the intermediate structure is approached the procedure is applicable to assembled structures that are either positive definite or positive semidefinite moreover the procedure represents an extension of the concept of subspace iteration to structures composed of substructures

quadratic forms on graphs we introduce a new graph parameter called the grothendieck constant of a graph g v e which is defined as the least constant k such that for every a e sup f v to s v 1 sum u v in e a u v cdot langle f u f v rangle le k sup varphi v to 1 1 sum u v in e a u v cdot varphi u varphi v the classical grothendieck inequality corresponds to the case of bipartite graphs but the case of general graphs is shown to have various algorithmic applications indeed our work is motivated by the algorithmic problem of maximizing the quadratic form u v e a u v u v over all v 1 1 which arises in the study of correlation clustering and in the investigation of the spin glass model we give upper and lower estimates for the integrality gap of this program we show that the integrality gap is o log vartheta overline g where vartheta overline g is the lovasz theta function of the complement of g which is always smaller than the chromatic number of g this yields an efficient constant factor approximation algorithm for the above maximization problem for a wide range of graphs g we also show that the maximum possible integrality gap is always at least log g where g is the clique number of g in particular it follows that the maximum possible integrality gap for the complete graph on n vertices with no loops is logn more generally the maximum possible integrality gap for any perfect graph with chromatic number n is logn the lower bound for the complete graph improves a result of kashin and szarek on gram matrices of uniformly bounded functions and settles a problem of megretski and of charikar and wirth

new algorithm for the ising problem partition function for finite lattice graphs we present a new efficient method to find the ising problem partition function for finite lattice graphs embeddable on an arbitrary orientable surface with integral coupling constants bounded in the absolute value by a polynomial of the size of the lattice graph the algorithm has been implemented for toroidal lattices using modular arithmetic and the generalized nested dissection method the implementation has substantially better performance than any other as far as we know

on the cut polytope the cut polytopep c g of a graphg v e is the convex hull of the incidence vectors of all edge sets of cuts ofg we show some classes of facet defining inequalities ofp c g we describe three methods with which new facet defining inequalities ofp c g can be constructed from known ones in particular we show that inequalities associated with chordless cycles define facets of this polytope moreover for these inequalities a polynomial algorithm to solve the separation problem is presented we characterize the facet defining inequalities ofp c g ifg is not contractible tok 5 we give a simple characterization of adjacency inp c g and prove that for complete graphs this polytope has diameter one and thatp c g has the hirsch property a relationship betweenp c g and the convex hull of incidence vectors of balancing edge sets of a signed graph is studied

justifying and generalizing contrastive divergence we study an expansion of the log likelihood in undirected graphical models such as the restricted boltzmann machine rbm where each term in the expansion is associated with a sample in a gibbs chain alternating between two random variables the visible vector and the hidden vector in rbms we are particularly interested in estimators of the gradient of the log likelihood obtained through this expansion we show that its residual term converges to zero justifying the use of a truncation running only a short gibbs chain which is the main idea behind the contrastive divergence cd estimator of the log likelihood gradient by truncating even more we obtain a stochastic reconstruction error related through a mean field approximation to the reconstruction error often used to train autoassociators and stacked autoassociators the derivation is not specific to the particular parametric forms used in rbms and requires only convergence of the gibbs chain we present theoretical and empirical evidence linking the number of gibbs steps k and the magnitude of the rbm parameters to the bias in the cd estimator these experiments also suggest that the sign of the cd estimator is correct most of the time even when the bias is large so that cd k is a good descent direction even for small k

computational complexity of the ground state determination of atomic clusters the authors prove that determining the ground state of a cluster of identical atoms interacting under two body central forces belongs to the class of np hard problems this means that as yet no polynomial time algorithm solving this problem is known and moreover that it is very unlikely that such an algorithm exists it also suggests the need for good heuristics

using singular value decomposition of component eigenmodes for interface reduction the aim of this paper is to describe the development of a reduced order model for modal analysis in a design context the design process of most industrial systems is based on the re utilization of certain components here we propose a reduction method involving component eigenmodes to recover the modal behaviour of an assembled structure the contribution of this work is that it uses component eigenmodes to build an interface reduction basis lastly the reduction methodology proposed is compared to the craig and bampton method by applying it to two case studies of which one is an industrial model of an open rotor blade

theory of a systematic computational error in free energy differences systematic inaccuracy is inherent in any computational estimate of a nonlinear average due to the availability of only a finite number of data values n free energy differences delta f between two states or systems are critically important examples of such averages previous work has demonstrated empirically that the finite sampling error can be very large many times k b t in delta f estimates for simple molecular systems here we present a theoretical description of the inaccuracy including the exact solution of a sample problem the precise asymptotic behavior in terms of 1 n for large n the identification of a universal law and numerical illustrations the theory relies on corrections to the central and other limit theorems

the geometry of algorithms using hierarchical tensors in this paper the differential geometry of the novel hierarchical tucker format for tensors is derived the set ht k of tensors with fixed tree t and hierarchical rank k is shown to be a smooth quotient manifold namely the set of orbits of a lie group action corresponding to the non unique basis representation of these hierarchical tensors explicit characterizations of the quotient manifold its tangent space and the tangent space of ht k are derived suitable for high dimensional problems the usefulness of a complete geometric description is demonstrated by two typical applications first new convergence results for the nonlinear gauss seidel method on ht k are given notably and in contrast to earlier works on this subject the task of minimizing the rayleigh quotient is also addressed second evolution equations for dynamic tensor approximation are formulated in terms of an explicit projection operator onto the tangent space of ht k in addition a numerical comparison is made between this dynamical approach and the standard one based on truncated singular value decompositions 2013 elsevier inc

strongly elliptic boundary integral equations for electromagnetic transmission problems we study a boundary integral equation method for transmission problems for strongly elliptic differential operators which yields a strongly elliptic system of pseudodifferential operators and which therefore can be used for numerical computations with galerkin s procedure the method is shown to work for the vector helmholtz equation in 3 with electromagnetic transmission conditions we propose a slightly modified system of boundary values in order for the corresponding bilinear form to be coercive over h 1 we analyse the boundary integral equations using the calculus of pseudodifferential operators here the concept of the principal symbol is used to derive existence and regularity results for the solution

learning factor graphs in polynomial time and sample complexity we study the computational and sample complexity of parameter and structure learning in graphical models our main result shows that the class of factor graphs with bounded degree can be learned in polynomial time and from a polynomial number of training examples assuming that the data is generated by a network in this class this result covers both parameter estimation for a known network structure and structure learning it implies as a corollary that we can learn factor graphs for both bayesian networks and markov networks of bounded degree in polynomial time and sample complexity importantly unlike standard maximum likelihood estimation algorithms our method does not require inference in the underlying network and so applies to networks where inference is intractable we also show that the error of our learned model degrades gracefully when the generating distribution is not a member of the target class of networks in addition to our main result we show that the sample complexity of parameter learning in graphical models has an o 1 dependence on the number of variables in the model when using the kl divergence normalized by the number of variables as the performance criterion

dynamic substructure response abstract dynamic substructure methods enable one to analyze a large scale dynamic system by considering its constituents individually and the combined action afterwards steady state harmonic oscillation has been studied almost exclusively the methods are extended to general non steady state excitations the orthonormality condition of the modal vectors in terms of the masters is introduced a mixed mass matrix for initial conditions is derived the dynamic flexibility is expressed in spectral form for response calculation modal participation factors for both distributed and concentrated loads are given explicitly

analysis of the statistical behavior of the genetic cluster exact approximation abstract the genetic cluster exact approximation algorithm is an efficient method to calculate ground states of ea spin glasses the method can be used to study ground state landscapes by calculating many independent ground states for each realization of the disorder the algorithm is analyzed with respect to the statistics of the ground states and the valleys of the energy landscape furthermore the distribution inside each valley is evaluated it is shown that the algorithm does not lead to a true t 0 thermodynamic distribution i e each ground state has not the same frequency of occurrence when performing many runs an extension of the technique is outlined which guarantees that each ground states occurs with the same probability

exact determination of all ground states of random field systems in polynomial time abstract an algorithm is developed which allows calculating all ground states of ferromagnetic and unfrustrated antiferromagnetic ising systems with arbitrary site dependent fields by transforming the system into an equivalent network and calculating the maximal flow by a trial and error scheme a minimum cut is constructed which corresponds to a spin configuration in this way each ground state is calculated with a finite probability the algorithm is applied to site diluted antiferromagnets in external magnetic fields it is found that in this case its time complexity is approximately quadratic in the lattice size as an application we calculate the distribution of overlaps between ground states of the site diluted antiferromagnet in a strong magnetic field and we analyse the fractal structure of these ground states

correlation polytopes their geometry and complexity a family of polytopes correlation polytopes which arise naturally in the theory of probability and propositional logic is defined these polytopes are tightly connected to combinatorial problems in the foundations of quantum mechanics and to the ising spin model correlation polytopes exhibit a great deal of symmetry exponential size symmetry groups which leave the polytope invariant and act transitively on its vertices are defined using the symmetries a large family of facets is determined a conjecture concerning the full facet structure of correlation polytopes is formulated the conjecture however implies that np co np various complexity results are proved it is shown that deciding membership in a correlation polytope is an np complete problem and deciding facets is probably not even in np the relations between the polytope symmetries and its complexity are indicated

general purpose computation with neural networks a survey of complexity theoretic results we survey and summarize the literature on the computational aspects of neural network models by presenting a detailed taxonomy of the various models according to their complexity theoretic characteristics the criteria of classification include the architecture of the network feedforward versus recurrent time model discrete versus continuous state type binary versus analog weight constraints symmetric versus asymmetric network size finite nets versus infinite families and computation type deterministic versus probabilistic among others the underlying results concerning the computational power and complexity issues of perceptron radial basis function winner take all and spiking neural networks are briefly surveyed with pointers to the relevant literature in our survey we focus mainly on the digital computation whose inputs and outputs are binary in nature although their values are quite often encoded as analog neuron states we omit the important learning issues

how to center deep boltzmann machines this work analyzes centered restricted boltzmann machines rbms and centered deep boltzmann machines dbms where centering is done by subtracting offset values from visible and hidden variables we show analytically that i centered and normal boltzmann machines bms and thus rbms and dbms are different parameterizations of the same model class such that any normal bm rbm dbm can be transformed to an equivalent centered bm rbm dbm and vice versa and that this equivalence generalizes to artificial neural networks in general ii the expected performance of centered binary bms rbms dbms is invariant under simultaneous flip of data and offsets for any off set value in the range of zero to one iii centering can be reformulated as a different update rule for normal bms rbms dbms and iv using the enhanced gradient is equivalent to setting the offset values to the average over model and data mean furthermore we present numerical simulations suggesting that i optimal generative performance is achieved by subtracting mean values from visible as well as hidden variables ii centered binary rbms dbms reach significantly higher log likelihood values than normal binary rbms dbms iii centering variants whose offsets depend on the model mean like the enhanced gradient suffer from severe divergence problems iv learning is stabilized if an exponentially moving average over the batch means is used for the offset values instead of the current batch mean which also prevents the enhanced gradient from severe divergence v on a similar level of log likelihood values centered binary rbms dbms have smaller weights and bigger bias parameters than normal binary rbms dbms vi centering leads to an update direction that is closer to the natural gradient which is extremely efficient for training as we show for small binary rbms vii centering eliminates the need for greedy layer wise pre training of dbms which often even deteriorates the results independently of whether centering is used or not and ix centering is also beneficial for auto encoders

on products and line graphs of signed graphs their eigenvalues and energy abstract in this article we examine the adjacency and laplacian matrices and their eigenvalues and energies of the general product non complete extended p sum or neps of signed graphs we express the adjacency matrix of the product in terms of the kronecker matrix product and the eigenvalues and energy of the product in terms of those of the factor graphs for the cartesian product we characterize balance and compute expressions for the laplacian eigenvalues and laplacian energy we give exact results for those signed planar cylindrical and toroidal grids which are cartesian products of signed paths and cycles we also treat the eigenvalues and energy of the line graphs of signed graphs and the laplacian eigenvalues and laplacian energy in the regular case with application to the line graphs of signed grids that are cartesian products and to the line graphs of all positive and all negative complete graphs

on network bipartivity systems with two types of agents with a preference for heterophilous interaction produces networks that are more or less close to bipartite we propose two measures quantifying the notion of bipartivity the two measures one well known and natural but computationally intractable one computationally less complex but also less intuitive are examined on model networks that continuously interpolates between bipartite graphs and graphs with many odd circuits we find that the bipartivity measures increase as we tune the control parameters of the test networks to intuitively increase the bipartivity and thus conclude that the measures are quite relevant we also measure and discuss the values of our bipartivity measures for empirical social networks constructed from professional collaborations internet communities and field surveys here we find as expected that networks arising from romantic online interaction have high and professional collaboration networks have low bipartivity values in some other cases probably due to low average degree of the network the bipartivity measures cannot distinguish between romantic and friendship oriented interaction

theory of minimum spanning trees i mean field theory and strongly disordered spin glass model the minimum spanning tree mst is a combinatorial optimization problem given a connected graph with a real weight cost on each edge find the spanning tree that minimizes the sum of the total cost of the occupied edges we consider the random mst in which the edge costs are quenched independent random variables there is a strongly disordered spin glass model due to newman and stein phys rev lett 72 2286 1994 which maps precisely onto the random mst we study scaling properties of random msts using a relation between kruskal s greedy algorithm for finding the mst and bond percolation we solve the random mst problem on the bethe lattice bl with appropriate wired boundary conditions and calculate the fractal dimension d 6 of the connected components viewed as a mean field theory the result implies that on a lattice in euclidean space of dimension d there are of order w d ensuremath d large connected components of the random mst inside a window of size w and that d d c d 6 is a critical dimension this differs from the value 8 suggested by newman and stein we also critique the original argument for 8 and provide an improved scaling argument that again yields d c 6 the result implies that the strongly disordered spin glass model has many ground states for dg6 and only of order one below six the results for msts also apply on the poisson weighted infinite tree which is a mean field approach to the continuum model of msts in euclidean space and is a limit of the bl in a companion paper we develop an ensuremath epsilon 6 ensuremath d expansion for the random mst on critical percolation clusters

minimizing thermodynamic length to select intermediate states for free energy calculations and replica exchange simulations in computational thermodynamics a sequence of intermediate states is commonly introduced to connect two equilibrium states we consider two cases where the choice of intermediate states is particularly important minimizing statistical error in free energy difference calculations and maximizing average acceptance probabilities in replica exchange simulations we derive bounds for these quantities in terms of the thermodynamic distance between the intermediates and show that in both cases the intermediates should be chosen as equidistant points along a geodesic connecting the end states

distribution of work in isothermal nonequilibrium processes diffusive motion in an externally driven potential is considered it is shown that the distribution of work required to drive the system from an initial equilibrium state to another is gaussian for slow but finite driving our result is obtained by projection method techniques exploiting a small parameter defined as the switching rate between the two states of the system the exact solution for a simple model system shows that such an expansion may fail in higher orders since the mean and the variance following from the exact distribution show nonanalytic behavior

grothendieck type inequalities in combinatorial optimization we survey connections of the grothendieck inequality and its variants to combinatorial optimization and computational complexity 2011 wiley periodicals inc

the computational complexity of pattern formation the computational complexity of diffusion limited aggregation and fluid invasion in porous media is studied the time requirements on an idealized parallel computer for simulating the patterns formed by these models are investigated it is shown that these growth models are p complete these results provide strong evidence that such pattern formation processes are inherently sequential and cannot be simulated efficiently in parallel

on the approximability of the energy function of ising spin glasses we consider polynomial time algorithms for finding approximate solutions to the ground state problem for the following three dimensional case of an ising spin glass n spins are arranged on a two level grid with n vertical interactions the main results are i there is an approximate polynomial time algorithm with absolute error less than n for all n and ii there exists a constant alpha 0 such that every approximate polynomial time algorithm has absolute error greater than alpha square root n infinitely often unless p np

approximate inference on planar graphs using loop calculus and belief propagation we introduce novel results for approximate inference on planar graphical models using the loop calculus framework the loop calculus chertkov and chernyak 2006a allows to express the exact partition function of a graphical model as a finite sum of terms that can be evaluated once the belief propagation bp solution is known in general full summation over all correction terms is intractable we develop an algorithm for the approach presented in chertkov et al 2008 which represents an efficient truncation scheme on planar graphs and a new representation of the series in terms of pfaffians of matrices we analyze the performance of the algorithm for models with binary variables and pairwise interactions on grids and other planar graphs we study in detail both the loop series and the equivalent pfaffian series and show that the first term of the pfaffian series for the general intractable planar model can provide very accurate approximations the algorithm outperforms previous truncation schemes of the loop series and is competitive with other state of the art methods for approximate inference

the computational complexity of generating random fractals we examine a number of models that generate random fractals the models are studied using the tools of computational complexity theory from the perspective of parallel computation diffusion limited aggregation and several widely used algorithms for equilibrating the ising model are shown to be highly sequential it is unlikely they can be simulated efficiently in parallel this is in contrast to mandelbrot percolation which can be simulated in constant parallel time our research helps shed light on the intrinsic complexity of these models relative to each other and to different growth processes that have been recently studied using complexity theory in addition the results may serve as a guide to simulation physics

stochastic relaxation gibbs distributions and the bayesian restoration of images we make an analogy between images and statistical mechanics systems pixel gray levels and the presence and orientation of edges are viewed as states of atoms or molecules in a lattice like physical system the assignment of an energy function in the physical system determines its gibbs distribution because of the gibbs distribution markov random field mrf equivalence this assignment also determines an mrf image model the energy function is a more convenient and natural mechanism for embodying picture attributes than are the local characteristics of the mrf for a range of degradation mechanisms including blurring nonlinear deformations and multiplicative or additive noise the posterior distribution is an mrf with a structure akin to the image model by the analogy the posterior distribution defines another imaginary physical system gradual temperature reduction in the physical system isolates low energy states annealing or what is the same thing the most probable states under the gibbs distribution the analogous operation under the posterior distribution yields the maximum a posteriori map estimate of the image given the degraded observations the result is a highly parallel relaxation algorithm for map estimation we establish convergence properties of the algorithm and we experiment with some simple pictures for which good restorations are obtained at low signal to noise ratios

making classical ground state spin computing fault tolerant we examine a model of classical deterministic computing in which the ground state of the classical system is a spatial history of the computation this model is relevant to quantum dot cellular automata as well as to recent universal adiabatic quantum computing constructions in its most primitive form systems constructed in this model cannot compute in an error free manner when working at nonzero temperature however by exploiting a mapping between the partition function for this model and probabilistic classical circuits we are able to show that it is possible to make this model effectively error free we achieve this by using techniques in fault tolerant classical computing and the result is that the system can compute effectively error free if the temperature is below a critical temperature we further link this model to computational complexity and show that a certain problem concerning finite temperature classical spin systems is complete for the complexity class merlin arthur this provides an interesting connection between the physical behavior of certain many body spin systems and computational complexity

solution procedures for three dimensional eddy current problems abstract the problem under consideration is that of the scattering of time periodic electromagnetic fields by metallic obstacles a common approximation here is that in which the metal is assumed to have infinite conductivity the resulting problem called the perfect conductor problem involves solving maxwell s equations in the region exterior to the obstacle with the tangential component of the electric field zero on the obstacle surface in the interface problem different sets of maxwell equations must be solved in the obstacle and outside while the tangential components of both electric and magnetic fields are continuous across the obstacle surface solution procedures for this problem are given there is an exact integral equation procedure for the interface problem and an asymptotic procedure for large conductivity both are based on a new integral equation procedure for the perfect conductor problem the asymptotic procedure gives an approximate solution by solving a sequence of problems analogous to the one for perfect conductors

equilibrium free energies from nonequilibrium measurements using maximum likelihood methods we present a maximum likelihood argument for the bennett acceptance ratio method and derive a simple formula for the variance of free energy estimates generated using this method this derivation of the acceptance ratio method using a form of logistic regression a common statistical technique allows us to shed additional light on the underlying physical and statistical properties of the method for example we demonstrate that the acceptance ratio method yields the lowest variance for any estimator of the free energy which is unbiased in the limit of large numbers of measurements

optimizing transformations of structural balance in signed networks with potential relationships abstract a signed network includes positive edges negative edges and 0 edges the last of which denote potential relationships however 0 edges are commonly ignored in transformations of unbalanced networks in this paper we take 0 edges into account and solve the optimization problem in a more comprehensive way we transform the unbalanced network by not only changing signs of edges but also changing edges into potential the experimental results show that our method can solve this problem efficiently and that our solutions are cost saving

bound entanglement in tree graphs in this paper we discuss the entanglement properties of graph diagonal states with particular emphasis on calculating the threshold for the transition between the presence and absence of entanglement i e the separability point special consideration is made of the thermal states of trees including the linear cluster state we characterize the type of entanglement present and describe the optimal entanglement witnesses and their implementation on a quantum computer up to an additive approximation in the case of general graphs we invoke a relation with the partition function of the classical ising model thereby intimating a connection to computational complexity theoretic tasks finally we show that the entanglement is robust to some classes of local perturbations

the complexity of counting self avoiding walks in subgraphs of two dimensional grids and hypercubes valiant siam j comput 8 1979 410 421 showed that the problem of computing the number of simple s t paths in graphs is p complete both in the case of directed graphs and in the case of undirected graphs welsh complexity knots colourings and counting cambridge university press cambridge 1993 p 17 asked whether the problem of computing the number of self avoiding walks of a given length in the complete two dimensional grid is complete for p1 the tally version of p this paper offers a partial answer to the question of welsh it is p complete to compute the number of self avoiding walks of a given length in a subgraph of a two dimensional grid several variations of the problem are also studied and shown to be p complete this paper also studies the problem of computing the number of self avoiding walks in a subgraph of a hypercube similar completeness results are shown for the problem by scaling the computation time to exponential it is shown that computing the number of self avoiding walks in hypercubes is a complete problem for exp in the case when a subgraph of a hypercube is specified by its dimension and a boolean circuit that accepts the nodes finally this paper studies the complexity of testing whether a given word over the four letter alphabet u d l r represents a self avoiding walk in a two dimensional grid a linear space lower bound is shown for nondeterministic turing machines with a 1 way input head to make this test

j ising model on durer lattices abstract this paper tackles the problem of finding analytical expressions describing the ground state properties of durer lattices over which a generalized edwards anderson model j ising model is defined a local frustration analysis is performed based on representative cells for these lattices the concentration of ferromagnetic f bonds x is used as the independent variable in the analysis 1 x is the concentration for antiferromagnetic a bonds where x spans the range 0 00 1 00 the presence of a bonds brings frustration whose clear manifestation is when bonds around the minimum possible circuit of bonds plaquette cannot be simultaneously satisfied the distribution of curved frustrated plaquettes within the representative cell is determinant for the evaluation of the parameters of interest such as average frustration segment energy per bond and fractional content of non frustrated bonds an analytical method is developed to cope with this analysis based on the direct probability of a plaquette being curved exact numerical simulations on a large number of randomly generated samples associated to durer lattices allow to validate previously described theoretical analysis this analytical method gives an excellent description for the most of the range for x a particular discussion for the point x 0 50 belonging to spin glass phase also sheds light on the general trends of the properties described here

model reductions for inference generality of pairwise binary and planar factor graphs we offer a solution to the problem of efficiently translating algorithms between different types of discrete statistical model we investigate the expressive power of three classes of model those with binary variables with pairwise factors and with planar topology as well as their four intersections we formalize a notion of simple reduction for the problem of inferring marginal probabilities and consider whether it is possible to simply reduce marginal inference from general discrete factor graphs to factor graphs in each of these seven subclasses we characterize the reducibility of each class showing in particular that the class of binary pairwise factor graphs is able to simply reduce only positive models we also exhibit a continuous spectral reduction based on polynomial interpolation which overcomes this limitation experiments assess the performance of standard approximate inference algorithms on the outputs of our reductions

tighter linear and semidefinite relaxations for max cut based on the lovasz schrijver lift and project procedure we study how the lift and project method introduced by lovasz and schrijver siam j optim 1 1991 pp 166 190 applies to the cut polytope we show that the cut polytope of a graph can be found in k iterations if there exist k edges whose contraction produces a graph with no k5 minor therefore for a graph g with n ge 4 nodes with stability number alpha g n 4 iterations suffice instead of the m number of edges iterations required in general and under some assumption n alpha g 3 iterations suffice the exact number of needed iterations is determined for small n le 7 by a detailed analysis of the new relaxations if positive semidefiniteness is added to the construction then one finds in one iteration a relaxation of the cut polytope which is tighter than its basic semidefinite relaxation and than another one introduced recently by anjos and wolkowicz discrete appl math to appear we also show how the lovasz schrijver relaxations for the stable set polytope of g can be strengthened using the corresponding relaxations for the cut polytope of the graph g nabla obtained from g by adding a node adjacent to all nodes of g

boundary integral equations for magnetic screens in ℝ 3 on developpe une methode d integrale limite pour la diffusion d ondes electromagnetiques sur des objets minces le probleme aux limites exterieur pour l equation de helmholtz vectorielle avec donnees de neumann sur un morceau de surface ouvert est converti en un systeme d equations integrales pour les sauts de la composante tangentielle du champ et sa divergence a travers l ecran une legere modification des donnees de cauchy donne un systeme fortement elliptique d equations pseudodifferentielles sur s

on the computational complexity of binary and analog symmetric hopfield nets we investigate the computational properties of finite binary and analog state discrete time symmetric hopfield nets for binary networks we obtain a simulation of convergent asymmetric networks by symmetric networks with only a linear increase in network size and computation time then we analyze the convergence time of hopfield nets in terms of the length of their bit representations here we construct an analog symmetric network whose convergence time exceeds the convergence time of any binary hopfield net with the same representation length further we prove that the min energy problem for analog hopfield nets is np hard and provide a polynomial time approximation algorithm for this problem in the case of binary nets finally we show that symmetric analog nets with an external clock are computationally turing universal

algorithms for estimating the partition function of restricted boltzmann machines abstract accurate estimates of the normalization constants partition functions of energy based probabilistic models markov random fields are highly important for example for assessing the performance of models monitoring training progress and conducting likelihood ratio tests several algorithms for estimating the partition function in relation to a reference distribution have been introduced including annealed importance sampling ais and bennett s acceptance ratio method bar however their conceptual similarities and differences have not been worked out so far and systematic comparisons of their behavior in practice have been missing we devise a unifying theoretical framework for these algorithms which comprises existing variants and suggests new approaches it is based on a generalized form of crooks equality linking the expectation over a distribution of samples generated by a transition operator to the expectation over the distribution induced by the reversed operator the framework covers different ways of generating samples such as parallel tempering and path sampling an empirical comparison revealed the differences between the methods when estimating the partition function of restricted boltzmann machines and ising models in our experiments bar using parallel tempering worked well with a small number of bridging distributions while path sampling based ais performed best when many bridging distributions were available because bar gave the overall best results we favor it over ais furthermore the experiments showed the importance of choosing a proper reference distribution

a physicist s approach to number partitioning the statistical physics approach to the number partioning problem a classical np hard problem is both simple and rewarding very basic notions and methods from statistical mechanics are enough to obtain analytical results for the phase boundary that separates the easy to solve from the hard to solve phase of the npp as well as for the probability distributions of the optimal and sub optimal solutions in addition it can be shown that solving a number portioning problem of size n to some extent corresponds to locating the minimum in an unsorted list of o 2n numbers considering this correspondence it is not surprising that known heuristics for the partitioning problem are not signi4cantly better than simple random search

constrained codes as networks of relations we address the well known problem of determining the capacity of constrained coding systems while the one dimensional case is well understood to the extent that there are techniques for rigorously deriving the exact capacity in contrast computing the exact capacity of a two dimensional constrained coding system is still an elusive research challenge the only known exception in the two dimensional case is an exact however not rigorous solution to the run length limited rll system on the hexagonal lattice furthermore only exponential time algorithms are known for the related problem of counting the exact number of constrained two dimensional information arrays we present the first known rigorous technique that yields an exact capacity of a two dimensional constrained coding system in addition we devise an efficient polynomial time algorithm for counting the exact number of constrained arrays of any given size our approach is a composition of a number of ideas and techniques describing the capacity problem as a solution to a counting problem in networks of relations graph theoretic tools originally developed in the field of statistical mechanics techniques for efficiently simulating quantum circuits as well as ideas from the theory related to the spectral distribution of toeplitz matrices using our technique we derive a closed form solution to the capacity related to the path cover constraint in a two dimensional triangular array the resulting calculated capacity is path cover is a generalization of the well known one dimensional rll constraint for which the capacity is known to be

bias variance decompositions for likelihood based estimators the bias variance decomposition of mean squared error is well understood and relatively straightforward in this note a similar simple decomposition is derived valid for any kind of error measure that when using the appropriate probability model can be derived from a kullback leibler divergence or log likelihood

the critical exponents of the two dimensional ising spin glass revisited exact ground state calculations and monte carlo simulations due to an incorrect generation of the bond configuration we obtained incorrect monte carlo results new simulation runs with a corrected and enlarged set of samples has led to the results shown in figures 1 and 2 which replace figures 3 and 4 of the original paper the best data collapse of xl we obtained see figure 1 for and of the binder ratio gl see figure 2 for figure 1 scaling plot of xl for different system sizes and temperatures the value of the correlation length exponent is v 1 5 thus the average value of the correlation length exponent now reads it should be noted that all other results of the paper where independently generated samples were used are not affected by these corrections figure 2 scaling plot of the binder ration gl we obtained the best data collapse for v 1 9 we note that the new estimate is much closer to the results 1 and 2 and is not compatible with the scaling prediction y 1 v

from near to eternity spin glass planting tiling puzzles and constraint satisfaction problems we present a methodology for generating ising hamiltonians of tunable complexity and with a priori known ground states based on a decomposition of the model graph into edge disjoint subgraphs the idea is illustrated with a spin glass model defined on a cubic lattice where subproblems whose couplers are restricted to the two values 1 1 are specified on unit cubes and are parametrized by their local degeneracy the construction is shown to be equivalent to a type of three dimensional constraint satisfaction problem known as the tiling puzzle by varying the proportions of subproblem types the hamiltonian can span a dramatic range of typical computational complexity from fairly easy to many orders of magnitude more difficult than prototypical bimodal and gaussian spin glasses in three space dimensions we corroborate this behavior via experiments with different algorithms and discuss generalizations and extensions to different types of graphs

training products of experts by minimizing contrastive divergence it is possible to combine multiple latent variable models of the same data by multiplying their probability distributions together and then renormalizing this way of combining individual expert models makes it hard to generate samples from the combined model but easy to infer the values of the latent variables of each expert because the combination rule ensures that the latent variables of different experts are conditionally independent when given the data a product of experts poe is therefore an interesting candidate for a perceptual system in which rapid inference is vital and generation is unnecessary training a poe by maximizing the likelihood of the data is difficult because it is hard even to approximate the derivatives of the renormalization term in the combination rule fortunately a poe can be trained using a different objective function called contrastive divergence whose derivatives with regard to the parameters can be approximated accurately and efficiently examples are presented of contrastive divergence learning using several types of expert on several types of data

fast approximate maximum a posteriori restoration of multicolour images we propose a new algorithm for the approximation of the maximum a posteriori map restoration of noisy images the image restoration problem is considered in a bayesian setting we assume as prior distribution multicolour markov random fields on a graph whose main restriction is the presence of only pairwise site interactions the noise is modelled as a bernoulli field computing the mode of the posterior distribution is np complete i e can very likely be done only in a time exponential in the number of sites of the underlying graph our algorithm runs in polynomial time and is based on the coding of the colours it produces an image with the following property either a pixel is coloured with one of the possible colours or it is left blank in the first case we prove that this is the colour of the site in the exact map restoration the quality of the approximation is then measured by the number of sites being left blank we assess the performance of the new algorithm by numerical experiments on the simple three colour potts model more rigorously we present a probabilistic analysis of the algorithm the results indicate that the approximation is quite often sufficiently good for the interpretation of the image

tensor network contractions for sat the computational cost of counting the number of solutions satisfying a boolean formula which is a problem instance of sat has proven subtle to quantify even when finding individual satisfying solutions is computationally easy e g 2 sat which is in mathsf p determining the number of solutions can be mathsf p hard recently computational methods simulating quantum systems experienced advancements due to the development of tensor network algorithms and associated quantum physics inspired techniques by these methods we give an algorithm using an axiomatic tensor contraction language for n variable sat instances with complexity o g cd o 1 2 c where c is the number of copy tensors g is the number of gates and d is the maximal degree of any copy tensor thus n variable counting problems can be solved efficiently when their tensor network expression has at most o log n copy tensors and polynomial fan out this framework also admits an intuitive proof of a variant of the tovey conjecture the r 1 sat instance of the dubois tovey theorem this study increases the theory expressiveness and application of tensor based algorithmic tools and provides an alternative insight on these problems which have a long history in statistical physics and computer science

quantum computational complexity of the n representability problem qma complete we study the computational complexity of the n representability problem in quantum chemistry we show that this problem is quantum merlin arthur complete which is the quantum generalization of nondeterministic polynomial time complete our proof uses a simple mapping from spin systems to fermionic systems as well as a convex optimization technique that reduces the problem of finding ground states to n representability

percolation of unsatisfiability in finite dimensions the optimization of two dimensional boolean formulas is studied using percolation theory rare region arguments and boundary effects in contrast with mean field results there is no satisfiability transition as the constraint density is varied although there is a logical connectivity transition in the disconnected phase there is a transition in the solution time the thermodynamic ground state for this np hard optimization problem is unique local solutions can be adjoined to find the global ground state these results have implications for the computational study of disordered materials

the ugc hardness threshold of the lp grothendieck problem for p 2 we consider the problem of given an n n matrix a aij whose diagonal entries vanish approximating in polynomial time the number rm opt p a max left sum i j 1 na ij x ix j x 1 ldots x n in mathbb r n wedge left sum i 1 n x i p right 1 p le 1 right when p 2 this is simply the problem of computing the maximum eigenvalue of a whereas for p actually it suffices to take p log n it is the grothendieck problem on the complete graph which was shown to have a o log n approximation algorithm in nemirovski et al nemirovski a c roos t terlaky 1999 on maximization of quadratic form over intersection of ellipsoids with common center math program ser a86 3 463 473 megretski megretski a 2001 relaxations of quadratic programs in operator theory and system analysis systems approximation singular integral operators and related topics bordeaux 2000 vol 129 operator theory advances and applications birkhauser basel 365 392 charikar and wirth charikar m a wirth 2004 maximizing quadratic programs extending grothendieck s inequality proc 45th annual sympos foundations comput sci ieee computer society 54 60 and was used in the work of charikar and wirth noted above to design the best known algorithm for the problem of computing the maximum correlation in correlation clustering thus the problem of approximating optp a interpolates between the spectral p 2 case and the correlation clustering p case from a physics point of view this problem corresponds to computing the ground states of spin glasses in a hard wall potential well we design a polynomial time algorithm which given p 2 and an n n matrix a aij with zeros on the diagonal computes optp a up to a factor p e 30 log p on the other hand assuming the unique games conjecture ugc we show that it is np hard to approximate optp a up to a factor smaller than p e hence as p the ugc hardness threshold for computing optp a is exactly p e 1 o 1

solving the fixed rank convex quadratic maximization in binary variables by a parallel zonotope construction algorithm abstract we address the weighted max cut problem or equivalently the problem of maximizing a quadratic form in n binary variables if the underlying symmetric matrix is positive semidefinite of fixed rank d then the problem can be reduced to searching the extreme points of a zonotope thus becoming of polynomial complexity in o n d 1 reverse search is an efficient and practical means for enumerating the cells of a regular hyperplane arrangement or equivalently the extreme points of a zonotope we present an enhanced version of reverse search of significantly reduced computational complexity that uses ray shooting and is well suited for parallel computation furthermore a neighborhood zonotope edge following descent heuristic can be devised we report preliminary computational experiments of a parallel implementation of our algorithms

polynomial time approximation schemes for max bisection on planar and geometric graphs the max bisection and min bisection problems are to find a partition of the vertices of a graph into two equal size subsets that respectively maximizes or minimizes the number of edges with endpoints in both subsets we design the first polynomial time approximation scheme for the max bisection problem on arbitrary planar graphs solving a long standing open problem the method of solution involves designing exact polynomial time algorithms for computing optimal partitions of bounded treewidth graphs in particular max and min bisection which could be of independent interest using a similar method we design also the first polynomial time approximation scheme for max bisection on unit disk graphs which could also be easily extended to other geometrically defined graphs

boundary integral equations for the scattering of electromagnetic waves by a homogeneous dielectric obstacle time harmonic electromagnetic waves are scattered by a homogeneous dielectric obstacle the corresponding electromagnetic transmission problem is reduced to a single integral equation over s for a single unknown tangential vector field where s is the interface between the obstacle and the surrounding medium in fact several different integral equations are derived and analysed including two previously known equations due to e marx and j r mautz and two new singular integral equations mautz s equation is shown to be uniquely solvable at all frequencies a new uniquely solvable singular integral equation is also found the paper also includes a review of methods using pairs of coupled integral equations over s it is these methods that are usually used in practice although single integral equations seem to offer some computational advantages

fast computing global structural balance in signed networks based on memetic algorithm abstract structural balance is a large area of study in signed networks and it is intrinsically a global property of the whole network computing global structural balance in signed networks which has attracted some attention in recent years is to measure how unbalanced a signed network is and it is a nondeterministic polynomial time hard problem many approaches are developed to compute global balance however the results obtained by them are partial and unsatisfactory in this study the computation of global structural balance is solved as an optimization problem by using the memetic algorithm the optimization algorithm named meme sb is proposed to optimize an evaluation function energy function which is used to compute a distance to exact balance our proposed algorithm combines genetic algorithm and a greedy strategy as the local search procedure experiments on social and biological networks show the excellent effectiveness and efficiency of the proposed method

the computational complexity of the self avoiding walk on random lattices it is proved that finding a minimum energy n step self avoiding walk on a 2d square lattice with assigned site energies is an np complete problem however if the fraction of lowest energy sites exceeds the percolation threshold then it is argued the problem can almost always be solved in polynomial time

on the approximability of the ground state problem for ising spin glasses by quantum devices a lower bound result about the approximability of the energy function of ising spin glasses in the quantum context is determined the ground state problem for the three dimensional ising spin glasses represented on two level grids such that the vertical interactions are at most p n where n is the number of spins set on the grid is considered we prove that unless np bqp there is a constant 0 such that every quantum approximate polynomial time algorithm finds a solution with absolute error greater than p n with high probability infinitely often

seizure warning algorithm based on optimization and nonlinear dynamics there is growing evidence that temporal lobe seizures are preceded by a preictal transition characterized by a gradual dynamical change from asymptomatic interictal state to seizure we herein report the first prospective analysis of the online automated algorithm for detecting the preictal transition in ongoing eeg signals such the algorithm constitutes a seizure warning system the algorithm estimates stlmax a measure of the order or disorder of the signal of eeg signals recorded from individual electrode sites the optimization techniques were employed to select critical brain electrode sites that exhibit the preictal transition for the warning of epileptic seizures specifically a quadratically constrained quadratic 0 1 programming problem is formulated to identify critical electrode sites the automated seizure warning algorithm was tested in continuous long term eeg recordings obtained from 5 patients with temporal lobe epilepsy for individual patient we use the first half of seizures to train the parameter settings which is evaluated by roc receiver operating characteristic curve analysis with the best parameter setting the algorithm applied to all cases predicted an average of 91 7 of seizures with an average false prediction rate of 0 196 per hour these results indicate that it may be possible to develop automated seizure warning devices for diagnostic and therapeutic purposes

rare events and the convergence of exponentially averaged work values equilibrium free energy differences are given by exponential averages of nonequilibrium work values such averages however often converge poorly as they are dominated by rare realizations i show that there is a simple and intuitively appealing description of these rare but dominant realizations this description is expressed as a duality between forward and reverse processes and provides both heuristic insights and quantitative estimates regarding the number of realizations needed for convergence of the exponential average analogous results apply to the equilibrium perturbation method of estimating free energy differences the pedagogical example of a piston and gas r c lua and a y grosberg j phys chem b 109 6805 2005 is used to illustrate the general discussion

complexity of tile rotation problems in this paper we introduce tile rotation problems the instances or initial configurations are tile assignments on a n x n lattice board and the question to be answered is the following does there exist any configuration obtained horn the initial one by tile rotations only whose cost is less than a given bound notice that a zero cost configuration corresponds to a perfect tiling we prove here the np completeness for both the zero cost problem for a particular set of 5 tiles and the minimization problem for a particular set of 2 tiles finally by showing the polynomiality of some subproblems we establish complexity border results

free energy reconstruction from steered dynamics without post processing various methods achieving importance sampling in ensembles of nonequilibrium trajectories enable one to estimate free energy differences and by maximum likelihood post processing to reconstruct free energy landscapes here based on bayes theorem we propose a more direct method in which a posterior likelihood function is used both to construct the steered dynamics and to infer the contribution to equilibrium of all the sampled states the method is implemented with two steering schedules first using non autonomous steering we calculate the migration barrier of the vacancy in email protected second using an autonomous scheduling related to metadynamics and equivalent to temperature accelerated molecular dynamics we accurately reconstruct the two dimensional free energy landscape of the 38 atom lennard jones cluster as a function of an orientational bond order parameter and energy down to the solid solid structural transition temperature of the cluster and without maximum likelihood post processing

j ising model on homogeneous archimedean lattices abstract we tackle the problem of finding analytical expressions describing the ground state properties of homogeneous archimedean lattices over which a generalized edwards anderson model j ising model is defined a local frustration analysis is performed based on representative cells for square lattices triangular lattices and honeycomb lattices the concentration of ferromagnetic f bonds x is used as the independent variable in the analysis 1 x is the concentration for antiferromagnetic a bonds where x spans the range 0 0 1 0 the presence of a bonds brings frustration whose clear manifestation is when bonds around the minimum possible circuit of bonds plaquette cannot be simultaneously satisfied the distribution of curved frustrated plaquettes within the representative cell is determinant for the evaluation of the parameters of interest such as average frustration segment energy per bond and fractional content of unfrustrated bonds two methods are developed to cope with this analysis one based on the direct probability of a plaquette being curved the other one is based on the consideration of the different ways bonds contribute to the particular plaquette configuration exact numerical simulations on a large number of randomly generated samples allow to validate previously described theoretical analysis it is found that the second method presents slight advantages over the first one however both methods give an excellent description for most of the range for x the small deviations at specific intervals of x for each lattice have to do with the self imposed limitations of both methods due to practical reasons a particular discussion for the point x 0 5 for each one of the lattices also shines light on the general trends of the properties described here

the influence of oppositely classified examples on the generalization complexity of boolean functions in this paper we analyze boolean functions using a recently proposed measure of their complexity this complexity measure motivated by the aim of relating the complexity of the functions with the generalization ability that can be obtained when the functions are implemented in feed forward neural networks is the sum of a number of components we concentrate on the case in which we use the first two of these components the first is related to the average sensitivity of the function and the second is in a sense a measure of the randomness or lack of structure of the function in this paper we investigate the importance of using the second term in the complexity measure and we consider to what extent these two terms suffice as an indicator of how difficult it is to learn a boolean function we also explore the existence of very complex boolean functions considering in particular the symmetric boolean functions

the swendsen wang process does not always mix rapidly the swendsen wang process provides one possible dynamics for the q state potts model computer simulations of this process are widely used to estimate the expectations of various observables random variables of a potts system in the equilibrium or gibbs distribution the legitimacy of such simulations depends on the rate of convergence of the process to equilibrium as measured by the mixing time empirical observations suggest that the mixing time of the swendsen wang process is short in many instances of practical interest although proofs of this desirable behavior are known only for some very special cases nevertheless we show that there are occasions when the mixing time of the swendsen wang process is exponential in the size of the system this undesirable behavior is related to the phenomenon of first order phase transitions in potts systems with q 2 states

more really is different abstract in 1972 p w anderson suggested that more is different meaning that complex physical systems may exhibit behavior that cannot be understood only in terms of the laws governing their microscopic constituents we strengthen this claim by proving that many macroscopic observable properties of a simple class of physical systems the infinite periodic ising lattice cannot in general be derived from a microscopic description this provides evidence that emergent behavior occurs in such systems and indicates that even if a theory of everything governing all microscopic interactions were discovered the understanding of macroscopic order is likely to require additional insights

complexity of two dimensional patterns in dynamical systems such as cellular automata and iterated maps it is often useful to look at a language or set of symbol sequences produced by the system there are well established classification schemes such as the chomsky hierarchy with which we can measure the complexity of these sets of sequences and thus the complexity of the systems which produce them in this paper we look at the first few levels of a hierarchy of complexity for two or more dimensional patterns we show that several definitions of regular language or local rule that are equivalent in d 1 lead to distinct classes in d 2 we explore the closure properties and computational complexity of these classes including undecidability and l nl and np completeness results we apply these classes to cellular automata in particular to their sets of fixed and periodic points finite time images and limit sets we show that it is undecidable whether a ca in d 2 has a periodic point of a given period and that certain local lattice languages are not finite time images or limit sets of any ca we also show that the entropy of a d dimensional ca s finite time image cannot decrease faster than t d unless it maps every initial condition to a single homogeneous state

optimization via enumeration a new algorithm for the max cut problem we present a polynomial time algorithm to find the maximum weight of an edge cut in graphs embeddable on an arbitrary orientable surface with integral weights bounded in the absolute value by a polynomial of the size of the graph

consistent inference of a general model using the pseudolikelihood method recently a maximum pseudolikelihood mpl inference method has been successfully applied to statistical physics models with intractable likelihoods we use information theory to derive a relation between the pseudolikelihood and likelihood functions furthermore we show the consistency of the pseudolikelihood method for a general model

exact ground states of large two dimensional planar ising spin glasses studying spin glass physics through analyzing their ground state properties has a long history although there exist polynomial time algorithms for the two dimensional planar case where the problem of finding ground states is transformed to a minimum weight perfect matching problem the reachable system sizes have been limited both by the needed cpu time and by memory requirements in this work we present an algorithm for the calculation of exact ground states for two dimensional ising spin glasses with free boundary conditions in at least one direction the algorithmic foundations of the method date back to the work of kasteleyn from the 1960s for computing the complete partition function of the ising model using kasteleyn cities we calculate exact ground states for huge two dimensional planar ising spin glass lattices up to 3000 2 spins within reasonable time according to our knowledge these are the largest sizes currently available kasteleyn cities were recently also used by thomas and middleton in the context of extended ground states on the torus moreover they show that the method can also be used for computing ground states of planar graphs furthermore we point out that the correctness of heuristically computed ground states can easily be verified finally we evaluate the solution quality of heuristic variants of the l bieche et al approach

a bound for the convergence rate of parallel tempering for sampling restricted boltzmann machines sampling from restricted boltzmann machines rbms is done by markov chain monte carlo mcmc methods the faster the convergence of the markov chain the more efficiently can high quality samples be obtained this is also important for robust training of rbms which usually relies on sampling parallel tempering pt an mcmc method that maintains several replicas of the original chain at higher temperatures has been successfully applied for rbm training we present the first analysis of the convergence rate of pt for sampling from binary rbms the resulting bound on the rate of convergence of the pt markov chain shows an exponential dependency on the size of one layer and the absolute values of the rbm parameters it is minimized by a uniform spacing of the inverse temperatures which is often used in practice similarly as in the derivation of bounds on the approximation error for contrastive divergence learning our bound on the mixing time implies an upper bound on the error of the gradient approximation when the method is used for rbm training

j ising model on mixed archimedean lattices 33 42 32 4 3 4 3 122 4 6 12 abstract this paper addresses the problem of finding analytical expressions describing the ground state properties of mixed archimedean lattices over which a generalized edwards anderson model j ising model is defined a local frustration analysis is performed based on representative cells for 3 3 4 2 3 2 4 3 4 3 12 2 and 4 6 12 lattices following the notation proposed by grunbaum and shephard the concentration of ferromagnetic f bonds x is used as the independent variable in the analysis 1 x is the concentration for antiferromagnetic a bonds where x spans the range 0 00 1 00 the presence of a bonds brings frustration whose clear manifestation is when bonds around the minimum possible circuit of bonds plaquette cannot be simultaneously satisfied the distribution of curved frustrated plaquettes within the representative cell is determinant for the evaluation of the parameters of interest such as average frustration segment energy per bond and fractional content of unfrustrated bonds two methods are developed to cope with this analysis one based on the direct probability of a plaquette being curved the other one is based on the consideration of the different ways bonds contribute to the particular plaquette configuration exact numerical simulations on a large number of randomly generated samples associated to 3 3 4 2 and 3 2 4 3 4 lattices allow to validate the previously described theoretical analysis it is found that the first method presents slight advantages over the second one however both methods give an excellent description for most of the range for x the small deviations at specific intervals of x for each lattice are relevant to the self imposed limitations of both methods due to practical reasons a particular discussion for the point x 0 50 for each one of the four lattices also sheds light on the general trends of the properties described here