graph optimisation problems and the potts glass the np complete problems of partitioning and colouring of random graphs with p partitions and colours respectively are mapped onto the statistical mechanical problem of p state potts glasses an estimate of the cost functions of these optimisation problems has been derived using the potts glass mean field theory this estimate applies to dense graphs in the thermodynamic limit an exact expression for the cost function in the large p limit is given

the statistical mechanics of the ising perceptron the authors investigate the problem of loading a single layered perceptron of ising or 1 weights focusing on the cases of linear and binary output neurons previous studies of this problem have been made using the canonical ensemble which in many cases require the breaking of replica symmetry they consider here an alternative approach based on the microcanonical ensemble and show that many results that were obtained previously using replica symmetry breaking within the canonical ensemble can be easily obtained using the replica symmetric assumption within the microcanonical ensemble since the nature of the replica symmetry breaking in many of the models with discrete weights is similar they believe that the results can immediately be extended to other cases such as learning a rule from examples and utilizing discrete weights of larger synaptic depth

statistical mechanics of the knapsack problem the knapsack problem is an np complete combinatorial optimization problem with inequality constraints using the replica method of statistical physics we study the space of its solutions for a large problem size it turns out that this problem is closely related to the theory of the binary perceptron

frozen states of a disordered globular heteropolymer the disordered heteropolymer chain with a random sequence of links a simple model of globular protein is investigated using the replica approach in the mean field approximation it is shown that for low dimensions of space d 1 it follows that the set of equilibrium states is ultrametric with an anomalously large number of weakly overlapping states

neural networks for optimization problems with inequality constraints the knapsack problem a strategy for finding approximate solutions to discrete optimization problems with inequality constraints using mean field neural networks is presented the constraints x d 0 are encoded by x x terms in the energy function a careful treatment of the mean field approximation for the self coupling parts of the energy is crucial and results in an essentially parameter free algorithm this methodology is extensively tested on the knapsack problem of size up to 103 items the algorithm scales like nm for problems with n items and m constraints comparisons are made with an exact branch and bound algorithm when this is computationally possible n d 30 the quality of the neural network solutions consistently lies above 95 of the optimal ones at a significantly lower cpu expense for the larger problem sizes the algorithm is compared with simulated annealing and a modified linear programming approach for nonhomogeneous problems these produce good solutions whereas for the more difficult homogeneous problems the neural approach is a winner with respect to solution quality and or cpu time consumption the approach is of course also applicable to other problems of similar structure like set covering

massey products and ideal class groups we consider certain massey products in the cohomology of a galois extension of fields with coefficients in p power roots of unity we prove formulas for these products both in general and in the special case that the galois extension in question is the maximal extension of a number field unramified outside a set of primes s including those above p and any archimedean places we then consider those zp kummer extensions l of the maximal p cyclotomic extension k of a number field k that are unramified outside s we show that massey products describe the structure of a certain decomposition free quotient of a graded piece of the maximal unramified abelian pro p extension of l in which all primes above those in s split completely with the grading arising from the augmentation filtration on the group ring of the galois group of l k we explicitly describe examples of the maximal unramified abelian pro p extensions of unramified outside p kummer extensions of the cyclotomic field of all p power roots of unity for irregular primes p

an efficient tabu search approach for the 0 1 multidimensional knapsack problem abstract in this paper we describe a new approach to tabu search ts based on strategic oscillation and surrogate constraint information that provides a balance between intensification and diversification strategies new rules needed to control the oscillation process are given for the 0 1 multidimensional knapsack 0 1 mkp based on a portfolio of test problems from the literature our method obtains solutions whose quality is at least as good as the best solutions obtained by previous methods especially with large scale instances these encouraging results confirm the efficiency of the tunneling concept coupled with surrogate information when resource constraints are present

an information based neural approach to generic constraint satisfaction a novel artificial neural network heuristic inn for general constraint satisfaction problems is presented extending a recently suggested method restricted to boolean variables in contrast to conventional ann methods it employs a particular type of non polynomial cost function based on the information balance between variables and constraints in a mean field setting implemented as an annealing algorithm the method is numerically explored on a testbed of graph coloring problems the performance is comparable to that of dedicated heuristics and clearly superior to that of conventional mean field annealing

polymers with long range self repulsion a variational approach the authors use a variational approach to study self interacting polymers with long range repulsion decaying for large distances as r lambda for 2 or lambda or 4 and lambda d where d is the spatial dimension they find that the exponent for the end to end distance of the chain is v 2 lambda with logarithmic corrections for lambda 2 or lambda 4 in agreement with renormalization group computations and recent numerical simulations they discuss the entire lambda d plane using a renormalization group near lambda 4 d 4

application of statistical mechanics to np complete problems in combinatorial optimisation recently developed techniques of the statistical mechanics of random systems are applied to the graph partitioning problem the averaged cost function is calculated and agrees well with numerical results the problem bears close resemblance to that of spin glasses the authors find a spin glass transition in the system and the low temperature phase space has an ultrametric structure this sheds light on the nature of hard computation problems

bootstrap method for some estimators based on fuzzy data abstract the bootstrap efron ann statist 7 1979 1 26 is a simple and straightforward method for calculating approximated biases standard deviations confidence intervals and so forth in almost any nonparametric estimation problem in this paper we consider another example namely fuzzy data and use the bootstrap to answer several questions concerning the minimum inaccuracy estimator corral and gil stochastica 8 1984 63 81 a what is the standard error of this estimator b what is a reasonable confidence interval for such an estimate the validity of bootstrap method is investigated using a real data and computer simulation

fluctuations in the random link matching problem using the replica approach and the cavity method we study the fluctuations of the optimal cost in the random link matching problem by means of replica arguments we derive the exact expression of its variance moreover we study the large deviation function deriving its expression in two different ways namely using both the replica method and the cavity method

an information based neural approach to constraint satisfaction a novel artificial neural network approach to constraint satisfaction problems is presented based on information theoretical considerations it differs from a conventional mean field approach in the form of the resulting free energy the method implemented as an annealing algorithm is numerically explored on a testbed of k sat problems the performance shows a dramatic improvement over that of a conventional mean field approach and is comparable to that of a state of the art dedicated heuristic gsat walk the real strength of the method however lies in its generality with minor modifications it is applicable to arbitrary types of discrete constraint satisfaction problems

a statistical mechanics analysis of the set covering problem the dependence of the optimal solution average cost of the set covering problem on the density of 1 s of the incidence matrix and on the number of constraints p is investigated in the limit where the number of items n goes to infinity the annealed approximation is employed to study two stochastic models the constant density model where the elements of the incidence matrix are statistically independent random variables and the karp model where the rows of the incidence matrix possess the same number of 1 s lower bounds for are presented in the case that p scales with ln n and is of order 1 as well as in the case that p scales linearly with n and is of order 1 n it is shown that in the case that p scales with exp n and is of order 1 the annealed approximation yields exact results for both models

explicit computation of galois p groups unramified at p abstract we introduce a new method for finding galois groups by modifying the p group generation algorithm using number theoretical information thus by exhaustive search we calculate the galois groups of some p extensions that were previously beyond reach

bayesian estimation in forest surveys when samples or prior information are fuzzy abstract fuzzy phenomena are frequently encountered in forestry due to the complexity of the systems under investigation in this paper existing inference methods are presented to accommodate situations in which random events take place in fuzzy circumstances fuzzy concepts are incorporated into these commonly accepted approaches to account for uncertainties used by factors other than randomness which traditional statistical approaches deal primarily with based on two extensions of likelihood functions for fuzzy samples bayesian estimates are generalized for use when sample information and prior distribution of parameters are fuzzy the estimators are developed for some typical membership functions for fuzzy samples and fuzzy priors applications are illustrated using examples for forestry

knapsack packing networks a knapsack packing neural network of 4n units with both low order and conjunctive asymmetric synapses is derived from a non hamiltonian energy function parallel simulations of randomly generated problems of size n in 5 10 20 are used to compare network solutions with those of simple greedy fast parallel enumerative algorithms

random walks with short range interaction and mean field behavior we introduce a model of self repelling random walks where the short range interaction between two elements of the chain decreases as a power of the difference in proper time the model interpolates between the lattice edwards model and the ordinary random walk we show by means of monte carlo simulations in two dimensions that the exponentvmf obtained through a mean field approximation correctly describes the numerical data and is probably exact as long as it is smaller than the corresponding exponent for self avoiding walks we also compute the exponent and present a numerical study of the scaling functions

adaptive and self averaging thouless anderson palmer mean field theory for probabilistic modeling we develop a generalization of the thouless anderson palmer tap mean field approach of disorder physics which makes the method applicable to the computation of approximate averages in probabilistic models for real data in contrast to the conventional tap approach where the knowledge of the distribution of couplings between the random variables is required our method adapts to the concrete set of couplings we show the significance of the approach in two ways our approach reproduces replica symmetric results for a wide class of toy models assuming a nonglassy phase with given disorder distributions in the thermodynamic limit on the other hand simulations on a real data model demonstrate that the method achieves more accurate predictions as compared to conventional tap approaches

when are networks truly modular abstract the study of cluster or community structure of complex networks contributes to the understanding of networks at a functional level in many networks latent classes of nodes are suspected which manifest themselves as communities i e groups of nodes with a high link density among the nodes of the same class and low link density between nodes of different classes community detection algorithms are used to infer these classes e g by finding a partition of the network which maximizes a quality function such as the network modularity q m newman m girvan finding and evaluating community structure in networks phys rev e 69 2004 026113 however it is known from numerical experiments that even purely random networks display intrinsic modularity and may be partitioned yielding high values of q extending on our earlier results j reichardt s bornholdt statistical mechanics of community detection phys rev e 74 2006 016110 the mapping of the community detection problem onto finding the ground state of a spin glass is exploited in order to derive analytical expressions for the expected modularity in random graphs and assess the theoretical limits to community detection the results are independent of any specific community detection algorithm and allow for differentiation between modularity arising purely due to the search process in the large configuration space of possible partitionings on the one hand or due to the actual presence of different classes of nodes on the other hand

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 with a one step replica symmetry breaking approximation the precise value of the critical average connectivity c q moreover we show that below c q there exists 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 leads us to propose a different algorithm that is able to color in polynomial time random graphs in the hard but colorable region i e when c in c d c q

interfaces in a random medium and replica symmetry breaking the authors develop a variational approach for studying interfaces and other manifolds in a disordered quenched medium the method may be applied to problems which range from directed polymers to the interface of an ising model in a random magnetic field they find that replica symmetry is spontaneously broken and the results of the flory approximation are recovered in a simple way corrections to this approximation may be computed in a systematic way

unramified subextensions of ray class field towers abstract fix a prime l in this paper we explore various senses in which the ramification in an infinite l extension of number fields can be small in particular 1 we construct infinitely many pairs of primes p q distinct from l such that q admits an infinite l extension unramified outside p q and 2 we explore the possibility of finding infinite unramified subextensions inside infinite ramified l extensions of number fields

heuristics for p class towers of imaginary quadratic fields cohen and lenstra have given a heuristic which for a fixed odd prime p leads to many interesting predictions about the distribution of p class groups of imaginary quadratic fields we extend the cohen lenstra heuristic to a non abelian setting by considering for each imaginary quadratic field k the galois group of the p class tower of k i e g k mathrm gal k infty k where k infty is the maximal unramified p extension of k by class field theory the maximal abelian quotient of g k is isomorphic to the p class group of k for integers c ge 1 we give a heuristic of cohen lenstra type for the maximal p class c quotient of g k and thereby give a conjectural formula for how frequently a given p group of p class c occurs in this manner in particular we predict that every finite schur sigma group occurs as g k for infinitely many fields k we present numerical data in support of these conjectures

the bayesian bootstrap the bayesian bootstrap is the bayesian analogue of the bootstrap instead of simulating the sampling distribution of a statistic estimating a parameter the bayesian bootstrap simulates the posterior distribution of the parameter operationally and inferentially the methods are quite similar because both methods of drawing inferences are based on somewhat peculiar model assumptions and the resulting inferences are generally sensitive to these assumptions neither method should be applied without some consideration of the reasonableness of these model assumptions in this sense neither method is a true bootstrap procedure yielding inferences unaided by external assumptions

statistical mechanics of the multi constraint continuous knapsack problem we apply the replica analysis established by gardner to the multi constraint continuous knapsack problem which is one of the linear programming problems and a most fundamental problem in the field of operations research or for a large problem size we analyse the space of solution and its volume and estimate the optimal number of items to go into the knapsack as a function of the number of constraints we study the stability of the replica symmetric rs solution and find that the rs calculation cannot estimate the optimal number of items in the knapsack correctly if many constraints are required in order to obtain a consistent solution in the rs region we try the zero entropy approximation for this continuous solution space and get a stable solution within the rs ansatz on the other hand in the replica symmetry breaking rsb region the one step rsb solution is found by parisi s scheme it turns out that this problem is closely related to the problem of optimal storage capacity and of generalization by maximum stability rule of a spherical perceptron

the fuzzy decision problem an approach to the point estimation problem with fuzzy information abstract this paper is devoted to the extension of the bayesian method for the point estimation when the available information is vague in the nonfuzzy case the parametric estimation can be approached as a particularization in the statistical decision problem this motivates us to accomplish the mentioned extension by looking at the parametric estimation in the fuzzy case as a special situation in the fuzzy decision problem defined by tanaka okuda and asia in this way concepts in the fuzzy decision problem are first expressed in the estimation terminology then on the basis of these concepts we shall introduce some notions and state some interesting results finally several illustrative examples will be exposed

an efficient mean field approach to the set covering problem a mean field feedback artificial neural network ann algorithm is developed and explored for the set covering problem a convenient encoding of the inequality constraints is achieved by means of a multilinear penalty function an approximate energy minimum is obtained by iterating a set of mean field equations in combination with annealing the approach is numerically tested against a set of publicly available test problems with sizes ranging up to 5 103 rows and 106 columns when comparing the performance with exact results for sizes where these are available the approach yields results within a few percent from the optimal solutions comparisons with other approximate methods also come out well in particular given the very low cpu consumption required typically a few seconds arbitrary problems can be processed using the algorithm via a public domain server 2001 elsevier science b v less

maximal 2 extensions with restricted ramification abstract we compute the galois groups of several 2 extensions of q ramified at finitely many odd primes

design of recurrent neural networks for solving constrained least absolute deviation problems recurrent neural networks for solving constrained least absolute deviation lad problems or l1 norm optimization problems have attracted much interest in recent years but so far most neural networks can only deal with some special linear constraints efficiently in this paper two neural networks are proposed for solving lad problems with various linear constraints including equality two sided inequality and bound constraints when tailored to solve some special cases of lad problems in which not all types of constraints are present the two networks can yield simpler architectures than most existing ones in the literature in particular for solving problems with both equality and one sided inequality constraints another network is invented all of the networks proposed in this paper are rigorously shown to be capable of solving the corresponding problems the different networks designed for solving the same types of problems possess the same structural complexity which is due to the fact these architectures share the same computing blocks and only differ in connections between some blocks by this means some flexibility for circuits realization is provided numerical simulations are carried out to illustrate the theoretical results and compare the convergence rates of the networks

extended hopfield models for combinatorial optimization the extended hopfield neural network proposed by abe et al 1992 for solving combinatorial optimization problems with equality and or inequality constraints has the drawback of being frequently stabilized in states with neurons of ambiguous classification as active or inactive we introduce in the model a competitive activation mechanism and we derive a new expression of the penalty energy allowing us to reduce significantly the number of neurons with intermediate level of activations the new version of the model is validated experimentally on the set covering problem our results confirm the importance of instituting competitive activation mechanisms in hopfield neural network models

solving optimization problems with mean field methods abstract a brief review is given for the use of feed back artificial neural networks ann to obtain good approximate solutions to combinatorial optimization problems the key element is the mean field approximation mft which differs from conventional methods and feels its ways towards good solutions rather than fully or partly exploring different possible solutions the methodology which is illustrated for the graphs bisection and knapsack problems is easily generalized to potts systems the latter is related to the deformable templates method which is illustrated with the track finding problem the mean field approximation is based on a variational principle which also turns out to be very profitable when computing correlations in polymers

a statistical analysis of the knapsack problem we investigate the dependence of the multi knapsack objective function on the knapsack capacities and on the number of capacity constraints p in the case when all n objects are assigned the same profit value and the weights are uniformly distributed over the unit interval a rigorous upper bound to the optimal profit is obtained employing the annealed approximation and then compared with the exact value obtained through the lagrangian relaxation method the analysis is restricted to the regime where n goes to infinity and p remains finite

a group theoretic characterization of linear groups let r be a group when is f linear this is an old problem the first to study this question systematically was malcev in 1940 m who essentially reduced the problem to finitely generated groups note that a finitely generated group is linear over some field of characteristic zero if and only if it can be embedded in cl c for some n very little progress was made since that paper of malcev although as linear groups are a quite special type of group many necessary conditions were obtained e g r should be residually finite and even virtually residually p for almost all primes p f should be virtually torsion free and if not solvable by finite it has a free non abelian subgroup etc cf z of course none of these properties characterizes the finitely generated linear groups over in this paper we give such a characterization using the congruence structure of r first some definitions for a group h d h denotes the minimal number of generators for h

matching hierarchical structures using association graphs it is well known that the problem of matching two relational structures can be posed as an equivalent problem of finding a maximal clique in a derived association graph however it is not clear how to apply this approach to computer vision problems where the graphs are hierarchically organized i e are trees since maximal cliques are not constrained to preserve the partial order we provide a solution to the problem of matching two trees by constructing the association graph using the graph theoretic concept of connectivity we prove that in the new formulation there is a one to one correspondence between maximal cliques and maximal subtree isomorphisms this allows us to cast the tree matching problem as an indefinite quadratic program using the motzkin straus theorem and we use replicator dynamical systems developed in theoretical biology to solve it such continuous solutions to discrete problems are attractive because they can motivate analog and biological implementations the framework is also extended to the matching of attributed trees by using weighted association graphs we illustrate the power of the approach by matching articulated and deformed shapes described by shock trees

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 exists a clustering phase c e 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

learning by dilution in a neural network a perceptron with n random weights can store of the order of n patterns by removing a fraction of the weights without changing their strengths the critical storage capacity as a function of the concentration of the remaining bonds for random outputs and for outputs given by a teacher perceptron is calculated a simple hebb like dilution algorithm is presented which in the teacher case reaches the optimal generalization ability

a novel optimizing network architecture with applications we present a novel optimizing network architecture with applications in vision learning pattern recognition and combinatorial optimization this architecture is constructed by combining the following techniques 1 deterministic annealing 2 self amplification 3 algebraic transformations 4 clocked objectives and 5 softassign deterministic annealing in conjunction with self amplification avoids poor local minima and ensures that a vertex of the hypercube is reached algebraic transformations and clocked objectives help partition the relaxation into distinct phases the problems considered have doubly stochastic matrix constraints or minor variations thereof we introduce a new technique softassign which is used to satisfy this constraint experimental results on different problems are presented and discussed

the multidimensional 0 1 knapsack problem an overview abstract the multidimensional 0 1 knapsack problem is one of the most well known integer programming problems and has received wide attention from the operational research community during the last four decades although recent advances have made possible the solution of medium size instances solving this np hard problem remains a very interesting challenge especially when the number of constraints increases this paper surveys the main results published in the literature the focus is on the theoretical properties as well as approximate or exact solutions of this special 0 1 program

graph optimization problems on a bethe lattice thep partitioning andp coloring problems on a bethe lattice of coordination numberz are analyzed it is shown that these two np complete optimization problems turn out to be equivalent to finding the ground state energy ofp state potts models with frustration numerical calculation of the cost function of both problems are carried out for several values ofz andp in the case ofp 2 the results are identical to those obtained by mezard and parisi for the case of the bipartitioning problem a numerical upper bound to the chromatic number is found for several values ofz

partitioning and modularity of graphs with arbitrary degree distribution we solve the graph bipartitioning problem in dense graphs with arbitrary degree distribution using the replica method we find the cut size to scale universally with k in contrast earlier results studying the problem in graphs with a poissonian degree distribution had found a scaling with kfu and anderson j phys a 19 1605 1986 our results also generalize to the problem of q partitioning they can be used to find the expected modularity q newman and girvan phys rev e 69 026113 2004 of random graphs and allow for the assessment of the statistical significance of the output of community detection algorithms given a graph or network gn m of n nodes and m edges the problem of graph partitioning is finding a partition of the nodes into q equal sized parts such that the number of edges connecting different parts the cut size is minimal the solution of this problem has many important practical applications in multiprocessor scheduling for parallel computing very large scale integrated chip design vlsi and data mining 1 due to the problem being np complete 2 i e no algorithm is known which is guaranteed to find an optimal solution in a number of steps that grows only polynomially with the size of the graph only heuristics exist to approximate a solution furthermore one cannot determine in polynomial time whether a solution found is indeed optimal the only way to assess the quality of a heuristic algorithm or a particular result is to compare with rigorous bounds on the cut size

on galois groups of unramified pro p extensions let p be an odd prime satisfying vandiver s conjecture we consider two objects the galois group x of the maximal unramified abelian pro p extension of the compositum of all z p extensions of q p and the galois group mathfrak g of the maximal unramified pro p extension of q mu p infty we give a lower bound for the height of the annihilator of x as an iwasawa module under some mild assumptions on bernoulli numbers we provide a necessary and sufficient condition for mathfrak g to be abelian the bound and the condition in the two results are given in terms of special values of a cup product pairing on cyclotomic p units we obtain in particular that for p 1 000 greenberg s conjecture that x is pseudo null holds and mathfrak g is in fact abelian

memorizing polymers shapes and permutations the authors discuss the equilibrium configurations of heteropolymers the interaction potential for each pair of monomers is chosen in such a way using a hebbian learning rule that the meta stable states of the polymer are some preassigned shapes the problem is analysed in some detail in the infinite dimensional case where the shapes correspond to permutations and the memory capacity is computed the one dimensional case is also considered

a potts neuron approach to communication routing a feedback neural network approach to communication routing problems is developed with emphasis on multiple shortest path problems with several requests for transmissions between distinct start and endnodes the basic ingredients are a set of potts neurons for each request with interactions designed to minimize path lengths and to prevent overloading of network arcs the topological nature of the problem is conveniently handled using a propagator matrix approach although the constraints are global the algorithmic steps are based entirely on local information facilitating distributed implementations in the polynomially solvable single request case the approach reduces to a fuzzy version of the bellman ford algorithm the method is evaluated for synthetic problems of varying sizes and load levels by comparing to exact solutions from a branch and bound method or to approximate solutions from a simple heuristic with very few exceptions the potts approach gives legal solutions of very high quality the computational demand scales merely as the product of the numbers of requests nodes and arcs

probabilistic analysis of the number partitioning problem given a sequence of n positive real numbers the number partitioning problem consists of partitioning them into two sets such that the absolute value of the difference of the sums of over the two sets is minimized in the case in which the s are statistically independent random variables uniformly distributed in the unit interval this np complete problem is equivalent to the problem of finding the ground state of an infinite range random antiferromagnetic ising model we employ the annealed approximation to derive analytical lower bounds to the average value of the difference for the best constrained and unconstrained partitions in the large n limit furthermore we calculate analytically the fraction of metastable states i e states that are stable against all single spin flips and found that it vanishes like