an algorithmic approach to the construction of homomorphisms induced by maps in homology this paper is devoted to giving the theoretical background for an algorithm for computing homomorphisms induced by maps in homology the principal idea is to insert the graph of a given continuous map f into ag raph of a multi valued representable map f the multi valued representable maps have well developed continuity properties and admit a nite coding that per mits treating them by combinatorial methods we provide the construction of the homomorphism f induced by f such that f f the presented construction does not require subsequent barycentric subdivisions and simpli cial approximations of f the main motivation for this paper comes from the project of computing the conley index for discrete dynamical systems

homology computation by reduction of chain complexes abstract a new algorithm for computing the homology module of a finitely generated chain complex is given it is based on local one step reductions of the size of the initial chain complex and it has a clear geometrical interpretation the complexity of the algorithm is discussed in special cases

arithmetic multivariate descartes rule let inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i be any number field or p adic field and consider f f 1 f k where f 1 f k inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 02i x 1 x n and no more than distinct exponent vectors occur in the monomial term expansions of the f i we prove that f has no more than 1 c n n 3 log n n geometrically isolated roots in inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 03i n where c is an explicit and effectively computable constant depending only on inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 04i this gives a significantly sharper arithmetic analogue of khovanski s theorem on real fewnomials and a higher dimensional generalization of an earlier result of hendrik w lenstra jr for the special case of a single univariate polynomial we also present some further refinements of our new bounds and a higher dimensional generalization of a bound of lipshitz on p adic complex roots connections to non archimedean amoebae and computational complexity including additive complexity and solving for the geometrically isolated rational roots are discussed along the way we thus provide the foundations for an effective arithmetic analogue of fewnomial theory

algorithmic pirogov sinai theory we develop an efficient algorithmic approach for approximate counting and sampling in the low temperature regime of a broad class of statistical physics models on finite subsets of the lattice mathbb z d and on the torus mathbb z n mathbb z d our approach is based on combining contour representations from pirogov sinai theory with barvinok s approach to approximate counting using truncated taylor series some consequences of our main results include an fptas for approximating the partition function of the hard core model at sufficiently high fugacity on subsets of mathbb z d with appropriate boundary conditions and an efficient sampling algorithm for the ferromagnetic potts model on the discrete torus mathbb z n mathbb z d at sufficiently low temperature

structured prediction dual extragradient and bregman projections we present a simple and scalable algorithm for maximum margin estimation of structured output models including an important class of markov networks and combinatorial models we formulate the estimation problem as a convex concave saddle point problem that allows us to use simple projection methods based on the dual extragradient algorithm nesterov 2003 the projection step can be solved using dynamic programming or combinatorial algorithms for min cost convex flow depending on the structure of the problem we show that this approach provides a memory efficient alternative to formulations based on reductions to a quadratic program qp we analyze the convergence of the method and present experiments on two very different structured prediction tasks 3d image segmentation and word alignment illustrating the favorable scaling properties of our algorithm

functional clones and expressibility of partition functions abstract we study functional clones which are sets of non negative pseudo boolean functions functions 0 1 k r 0 closed under essentially multiplication summation and limits functional clones naturally form a lattice under set inclusion and are closely related to counting constraint satisfaction problems csps we identify a sublattice of interesting functional clones and investigate the relationships and properties of the functional clones in this sublattice

bounding the convergence time of the gibbs sampler in bayesian image restoration agibbs mathstat yorku ca summary this paper shows how coupling methodology can be used to give precise a priori bounds on the convergence time of markov chain monte carlo algorithms for which a partial order exists on the state space which is preserved by the markov chain transitions this methodology is applied to give a bound on the convergence time of the random scan gibbs sampler used in the bayesian restoration of an image of n pixels for our algorithm in which only one pixel is updated at each iteration the bound is a constant times n2 the proportionality constant is given and is easily calculated these bounds also give an indication of the running time of coupling from the past algorithms

predicting critical crashes a new restriction for the free variables abstract several authors have noticed the signature of log periodic oscillations prior to large stock market crashes unfortunately the prediction of crashes with this method is difficult since good fits of the corresponding log periodic equation to stock market prices are also observed in quiet times we deduce a new inequality among the free variables in this equation that must be satisfied before a crash this inequality alone yields crash predictions with a 25 success rate for the dow jones index this compares to a 9 success rate obtained by previous approaches combining our method with previous ones we obtain a success rate of 54

the determination of canonical forms for lattice quadrature rules abstract lattice rules are equal weight numerical quadrature rules for the integration of periodic functions over the s dimensional unit hypercube us 0 1 s for a given lattice rule say ql a set of points l the integration lattice regularly spaced in all of rs is generated by a finite number of rational vectors the abscissa set for ql is then p ql l us it is known that p ql is a finite abelian group under addition modulo the integer lattice zs and that ql f may be written in the form of a nonrepetitive multiple sum q l f 1 n 1 n m j 1 1 n 1 j m 1 n m f j 1 n 1 z 1 j m n m z m known as a canonical form in which denotes addition modulo zs in this form zi zs m is called the rank and n1 n2 nm are called the invariants of ql and n i 1 n i for i 1 2 m 1 the rank and invariants are uniquely determined for a given lattice rule in this paper we provide a construction of a canonical form for a lattice rule ql given a generator set for the lattice l we then show how the rank and invariants of ql may be determined directly from the generators of the dual lattice l

efficient recursions for general factorisable models let n s valued categorical variables be jointly distributed according to a distribution known only up to an unknown normalising constant for an unnormalised joint likelihood expressible as a product of factors we give an algebraic recursion which can be used for computing the normalising constant and other summations a saving in computation is achieved when each factor contains a lagged subset of the components combining in the joint distribution with maximum computational efficiency as the subsets attain their minimum size if each subset contains at most r 1 of the n components in the joint distribution we term this a lag r model whose normalising constant can be computed using a forward recursion in o sr 1 computations as opposed to o sn for the direct computation we show how a lag r model represents a markov random field and allows a neighbourhood structure to be related to the unnormalised joint likelihood we illustrate the method by showing how the normalising constant of the ising or autologistic model can be computed

analyzing glauber dynamics by comparison of markov chains a popular technique for studying random properties of a combinatorial set is to design a markov chain monte carlo algorithm for many problems there are natural markov chains connecting the set of allowable configurations which are based on local moves or glauber dynamics typically these single site update algorithms are difficult to analyze so often the markov chain is modified to update several sites simultaneously recently there has been progress in analyzing these more complicated algorithms for several important combinatorial problems in this work we use the comparison technique of diaconis and saloff coste to show that several of the natural single point update algorithms are efficient the strategy is to relate the mixing rate of these algorithms to the corresponding nonlocal algorithms which have already been analyzed this allows us to give polynomial time bounds for single point update algorithms for problems such as generating planar tilings and random triangulations of convex polygons we also survey several other comparison techniques along with specific applications which have been used in the context of estimating mixing rates of markov chains

lppls bubble indicators over two centuries of the s p 500 index the aim of this paper is to present novel tests for the early causal diagnostic of positive and negative bubbles in the s p 500 index and the detection of end of bubble signals with their corresponding confidence levels we use monthly s p 500 data covering the period from august 1791 to august 2014 this study is the first work in the literature showing the possibility to develop reliable ex ante diagnostics of the frequent regime shifts over two centuries of data we show that the ds lppls log periodic power law singularity approach successfully diagnoses positive and negative bubbles constructs efficient end of bubble signals for all of the well documented bubbles and obtains for the first time new statistical evidence of bubbles for some other events we also compare the ds lppls method to the exponential curve fitting and the generalized sup adf test approaches and find that ds lppls system is more accurate in identifying well known bubble events with significantly smaller numbers of false negatives and false positives

glauber dynamics on trees and hyperbolic graphs we study continuous time glauber dynamics for random configurations with local constraints e g proper coloring ising and potts models on finite graphs with n vertices and of bounded degree we show that the relaxation time defined as the reciprocal of the spectral gap 1 2 for the dynamics on trees and on planar hyperbolic graphs is polynomial in n for these hyperbolic graphs this yields a general polynomial sampling algorithm for random configurations we then show that for general graphs if the relaxation time 2 satisfies 2 o 1 then the correlation coefficient and the mutual information between any local function which depends only on the configuration in a fixed window and the boundary conditions decays exponentially in the distance between the window and the boundary for the ising model on a regular tree this condition is sharp

high dimensional inference with random maximum a posteriori perturbations this paper presents a new approach called perturb max for high dimensional statistical inference in graphical models that is based on applying random perturbations followed by optimization this framework injects randomness into maximum a posteriori map predictors by randomly perturbing the potential function for the input a classic result from extreme value statistics asserts that perturb max operations generate unbiased samples from the gibbs distribution using high dimensional perturbations unfortunately the computational cost of generating so many high dimensional random variables can be prohibitive however when the perturbations are of low dimension sampling the perturb max prediction is as efficient as map optimization this paper shows that the expected value of perturb max inference with low dimensional perturbations can be used sequentially to generate unbiased samples from the gibbs distribution furthermore the expected value of the maximal perturbations is a natural bound on the entropy of such perturb max models a measure concentration result for perturb max values shows that the deviation of their sampled average from its expectation decays exponentially in the number of samples allowing effective approximation of the expectation

the ising partition function zeros and deterministic approximation we study the problem of approximating the partition function of the ferromagnetic ising model with both pairwise as well as higher order interactions equivalently in graphs as well as hypergraphs our approach is based on the classical lee yang theory of phase transitions along with a new lee yang theorem for the ising model with higher order interactions and on an extension of ideas developed recently by barvinok and patel and regts that can be seen as an algorithmic realization of the lee yang theory our first result is a deterministic polynomial time approximation scheme an fptas for the partition function in bounded degree graphs that is valid over the entire range of parameters beta the interaction and lambda the external field except for the case left lambda right 1 the zero field case a polynomial time randomized approximation scheme fpras for all graphs and all beta lambda based on markov chain monte carlo simulation has long been known unlike most other deterministic approximation algorithms for problems in statistical physics and counting our algorithm does not rely on the decay of correlations property but as pointed out above on lee yang theory this approach extends to the more general setting of the ising model on hypergraphs of bounded degree and edge size where no previous algorithms even randomized were known for a wide range of parameters in order to achieve this latter extension we establish a tight version of the lee yang theorem for the ising model on hypergraphs improving a classical result of suzuki and fisher

on the two dimensional dynamical ising model in the phase coexistence region we consider a glauber dynamics reversible with respect to the two dimensional ising model in a finite square of sidel in the absence of an external field and at large inverse temperature we first consider the gap in the spectrum of the generator of the dynamics in two different cases with plus and open boundary conditions we prove that when the symmetry under global spin flip is broken by the boundary conditions the gap is much larger than the case in which the symmetry is present for this latter we compute exactly the asymptotics of 1 l log gap asl and show that it coincides with the surface tension along one of the coordinate axes as a consequence we are able to study quite precisely the large deviations in time of the magnetization and to obtain an upper bound on the spin spin time correlation in the infinite volume plus phase our results establish a connection between the dynamical large deviations and those of the equilibrium gibbs measure studied by shlosman in the framework of the rigorous description of the wulff shape for the ising model finally we show that in the case of open boundary conditions it is possible to rescale the time withl in such a way that asl the finite dimensional distributions of the time rescaled magnetization converge to those of a symmetric continuous time markov chain on the two state space m m m being the spontaneous magnetization our methods rely upon a novel combination of techniques for bounding from below the gap of symmetric markov chains on complicated graphs developed by jerrum and sinclair in their markov chain approach to hard computational problems and the idea of introducing block glauber dynamics instead of the standard single site dynamics in order to put in evidence more effectively the effect of the boundary conditions in the approach to equilibrium

index of nilpotency of binomial ideals abstract in this paper we study and compute bounds for the index of nilpotency of lattice and cellular binomial ideals which are optimal in many cases this computations can be generalized to binomial ideals getting an effective hilbert s nullstellensatz for binomial ideals

ml parameter estimation for markov random fields with applications to bayesian tomography markov random fields mrfs have been widely used to model images in bayesian frameworks for image reconstruction and restoration typically these mrf models have parameters that allow the prior model to be adjusted for best performance however optimal estimation of these parameters sometimes referred to as hyperparameters is difficult in practice for two reasons i direct parameter estimation for mrfs is known to be mathematically and numerically challenging ii parameters can not be directly estimated because the true image cross section is unavailable we propose a computationally efficient scheme to address both these difficulties for a general class of mrf models and we derive specific methods of parameter estimation for the mrf model known as generalized gaussian mrf ggmrf we derive methods of direct estimation of scale and shape parameters for a general continuously valued mrf for the ggmrf case we show that the ml estimate of the scale parameter spl sigma has a simple closed form solution and we present an efficient scheme for computing the ml estimate of the shape parameter p by an off line numerical computation of the dependence of the partition function on p we present a fast algorithm for computing ml parameter estimates when the true image is unavailable to do this we use the expectation maximization em algorithm we develop a fast simulation method to replace the e step and a method to improve the parameter estimates when the simulations are terminated prior to convergence experimental results indicate that our fast algorithms substantially reduce the computation and result in good scale estimates for real tomographic data sets

approximating the partition function of the ferromagnetic potts model we provide evidence that it is computationally difficult to approximate the partition function of the ferromagnetic q state potts model when q 2 specifically we show that the partition function is hard for the complexity class rhpi under approximation preserving reducibility thus it is as hard to approximate the partition function as it is to find approximate solutions to a wide range of counting problems including that of determining the number of independent sets in a bipartite graph our proof exploits the first order phase transition of the random cluster model which is a probability distribution on graphs that is closely related to the q state potts model

a binomial approximation method for the ising model a large portion of the computation required for the partition function of the ising model can be captured with a simple formula in this work we support this claim by defining an approximation to the partition function and other thermodynamic quantities of the ising model that requires no algorithm at all this approximation which uses the high temperature expansion is solely based on the binomial distribution and performs very well at low temperatures at high temperatures we provide an alternative approximation which also serves as a lower bound on the partition function and is trivial to compute we provide theoretical evidence and the results of numerical experiments to support the strength of these approximations

recognizing badly presented z modules finitely generated z modules have canonical decompositions when such modules are given in a finitely presented form there is a classical algorithm for computing a canonical decomposition this is the algorithm for computing the smith normal form of an integer matrix we discuss algorithms for smith normal form computation and present practical algorithms which give excellent performance for modules arising from badly presented abelian groups we investigate such issues as congruential techniques sparsity considerations pivoting strategies for gauss jordan elimination lattice basis reduction and computational complexity our results which are primarily empirical show dramatically improved performance on previous methods

monte carlo methods for the ferromagnetic potts model using factor graph duality normal factor graph duality offers new possibilities for monte carlo algorithms in graphical models specifically we consider the problem of estimating the partition function of the ferromagnetic ising and potts models by monte carlo methods which are known to work well at high temperatures but to fail at low temperatures we propose monte carlo methods uniform sampling and importance sampling in the dual normal factor graph and demonstrate that they behave differently they work particularly well at low temperatures by comparing the relative error in estimating the partition function we show that the proposed importance sampling algorithm significantly outperforms the state of the art deterministic and monte carlo methods for the ferromagnetic ising model in an external field we show the equivalence between the valid configurations in the dual normal factor graph and the terms that appear in the high temperature series expansion of the partition function following this result we discuss connections with jerrum sinclair s polynomial randomized approximation scheme the subgraphs world process for evaluating the partition function of ferromagnetic ising models

universality of the mean field for the potts model we consider the potts model with q colors on a sequence of weighted graphs with adjacency matrices a n allowing for both positive and negative weights under a mild regularity condition on a n we show that the mean field prediction for the log partition function is asymptotically correct whenever mathrm tr a n 2 o n in particular our results are applicable for the ising and the potts models on any sequence of graphs with average degree going to infty using this we establish the universality of the limiting log partition function of the ferromagnetic potts model for a sequence of asymptotically regular graphs and that of the ising model for bi regular bipartite graphs in both ferromagnetic and anti ferromagnetic domain we also derive a large deviation principle for the empirical measure of the colors for the potts model on asymptotically regular graphs

algorithms for computing a hermite reduction of a matrix with polynomial coefficients resume nous etudions dans cet article des algorithmes de calcul de la reduction de hermite d une matrice a coefficients polynomiaux qui evitent l explosion de la taille des objets intermediaires et ont une bonne complexite sequentielle le deuxieme et le troisieme algorithmes generalisent la methode des sous resultants pour le calcul du pgcd de deux polynomes le dernier de ces algorithmes semble optimal en ce sens qu il calcule une reduction de hermite avec une matrice de passage de degre minimal en utilisant une methode progressive et economique nous ramenons le calcul de la reduite de hermite a un probleme de triangulation progressive des matrices dont les tailles sont bien controlees et dont les entrees sont des coefficients de polynomes donnes en entree l algorithme est donc faisable des qu on dispose d une methode de triangulation en temps polynomial pour les matrices dans le corps des coefficients il revient au meme de dire que le calcul des determinants est en temps polynomial dans le corps en question pour le codage choisi les resultats theorqiues sont meilleurs que pour les algorithmes connus precedemment et les resultats partiques sont encourageants

an improved lower bound for approximating minimum gcd multiplier in l norm gcdm in this paper we study the inapproximability of the following np complete number theoretic optimization problems introduced by rossner and seifert c rossner j p seifert the complexity of approximate optima for greatest common divisor computations in proceedings of the 2nd international algorithmic number theory symposium ants ii 1996 pp 307 322 given n numbers a 1 a n z find an minimum gcd multiplier for a 1 a n i e a vector x z n with minimum max 1 i n x i satisfying i 1 nx ia i gcd a 1 a n we show that assuming p np it is np hard to approximate the minimum gcd multiplier in norm gcdm within a factor n c l o g l o g n for some constant c 0 where n is the dimension of the given vector this improves on the best previous result the best result so far gave 2 l o g n 1 e factor hardness by rossner and seifert c rossner j p seifert the complexity of approximate optima for greatest common divisor computations in proceedings of the 2nd international algorithmic number theory symposium ants ii 1996 pp 307 322 where e 0 is an arbitrarily small constant

complexity classification of the six vertex model abstract we prove a complexity dichotomy theorem for the six vertex model for every setting of the parameters of the model we prove that computing the partition function is either solvable in polynomial time or p hard the dichotomy criterion is explicit

worst case complexity bounds on algorithms for computing the canonical structure of finite abelian groups and the hermite and smith normal forms of an integer matrix an o s 5 m s 2 algorithm for computing the canonical structure of a finite abelian group represented by an integer matrix of size s this is the smith normal form of the matrix is presented moreover an o s 3 m s 2 algorithm for computing the hermite normal form of an integer matrix of size s is given the upper bounds derived on the computational complexity of the algorithms above improve the upper bounds given by kannan and bachem in siam j comput 8 1979 pp 499 507 and chou and collins in siam j comput 11 1982 pp 687 708

the complexity of counting in sparse regular and planar graphs we show that a number of graph theoretic counting problems remain cal np hard indeed cal p complete in very restricted classes of graphs in particular we prove that the problems of counting matchings vertex covers independent sets and extremal variants of these all remain hard when restricted to planar bipartite graphs of bounded degree or regular graphs of constant degree we obtain corollaries about counting cliques in restricted classes of graphs and counting satisfying assignments to restricted classes of monotone 2 cnf formulae to achieve these results a new interpolation based reduction technique which preserves properties such as constant degree is introduced

counting in two spin models on d regular graphs supported in part by alfred p sloan research fellowship supported in part by department of defense ndseg fellowships

convergence rates of markov chains for some self assembly and non saturated ising models algorithms based on markov chains are ubiquitous across scientific disciplines as they provide a method for extracting statistical information about large complicated systems for some self assembly models markov chains can be used to predict both equilibrium and non equilibrium dynamics in fact the efficiency of these self assembly algorithms can be related to the rate at which simple chains converge to their stationary distribution we give an overview of the theory of markov chains and show how many natural chains including some relevant in the context of self assembly undergo a phase transition as a parameter representing temperature is varied in the model we illustrate this behavior for the non saturated ising model in which there are two types of tiles that prefer to be next to other tiles of the same type unlike the standard ising model we also allow empty spaces that are not occupied by either type of tile we prove that for a local markov chain that allows tiles to attach and detach from the lattice the rate of convergence is fast at high temperature and slow at low temperature

quantum speedup of monte carlo methods monte carlo methods use random sampling to estimate numerical quantities which are hard to compute deterministically one important example is the use in statistical physics of rapidly mixing markov chains to approximately compute partition functions in this work we describe a quantum algorithm which can accelerate monte carlo methods in a very general setting the algorithm estimates the expected output value of an arbitrary randomized or quantum subroutine with bounded variance achieving a near quadratic speedup over the best possible classical algorithm combining the algorithm with the use of quantum walks gives a quantum speedup of the fastest known classical algorithms with rigorous performance bounds for computing partition functions which use multiple stage markov chain monte carlo techniques the quantum algorithm can also be used to estimate the total variation distance between probability distributions efficiently

parallel algorithms for matrix normal forms abstract here we offer a new randomized parallel algorithm that determines the smith normal form of a matrix with entries being univariate polynomials with coefficients in an arbitrary field the algorithm has two important advantages over our previous one the multipliers relating the smith form to the input matrix are computed and the algorithm is probabilistic of las vegas type i e always finds the correct answer the smith form algorithm is also a good sequential algorithm our algorithm reduces the problem of smith form computation to two hermite form computations thus the smith form problem has complexity asymptotically that of the hermite form problem we also construct fast parallel algorithms for jordan normal form and testing similarity of matrices both the similarity and non similarity problems are in the complexity class rnc for the usual coefficient fields i e they can be probabilistically decided in polylogarithmic time using polynomially many processors

random walks totally unimodular matrices and a randomised dual simplex algorithm we discuss the application of random walks to generating a random basis of a totally unimodular matrix and to solving a linear program with such a constraint matrix we also derive polynomial upper bounds on the combinatorial diameter of an associated polyhedron

polynomial time approximation algorithms for the ising model the paper presents a randomised algorithm which evaluates the partition function of an arbitrary ferromagnetic ising system to any specified degree of accuracy the running time of the algorithm in

a stable and robust calibration scheme of the log periodic power law model abstract we present a simple transformation of the formulation of the log periodic power law formula of the johansen ledoit sornette jls model of financial bubbles that reduces it to a function of only three nonlinear parameters the transformation significantly decreases the complexity of the fitting procedure and improves its stability tremendously because the modified cost function is now characterized by good smooth properties with in general a single minimum in the case where the model is appropriate to the empirical data we complement the approach with an additional subordination procedure that slaves two of the nonlinear parameters to the most crucial nonlinear parameter the critical time t c defined in the jls model as the end of the bubble and the most probable time for a crash to occur this further decreases the complexity of the search and provides an intuitive representation of the results of the calibration with our proposed methodology metaheuristic searches are not longer necessary and one can resort solely to rigorous controlled local search algorithms leading to a dramatic increase in efficiency empirical tests on the shanghai composite index sse from january 2007 to march 2008 illustrate our findings

a chernoff bound for random walks on expander graphs we consider a finite random walk on a weighted graph g we show that the fraction of time spent in a set of vertices a converges to the stationary probability pi a with error probability exponentially small in the length of the random walk and the square of the size of the deviation from pi a the exponential bound is in terms of the expansion of g and improves previous results of d aldous probab engrg inform sci 1 1987 pp 33 46 l lovasz and m simonovits it random structures algorithms 4 1993 pp 359 412 m ajtai j komlos and e szemeredi deterministic simulation of logspace in proc 19th acm symp on theory of computing 1987 we show that taking the sample average from one trajectory gives a more efficient estimate of pi a than the standard method of generating independent sample points from several trajectories using this more efficient sampling method we improve the algorithms of jerrum and sinclair for approximating the number of perfect matchings in a dense graph and for approximating the partition function of a ferromagnetic ising system and we give an efficient algorithm to estimate the entropy of a random walk on an unweighted graph

counting over non planar graphs abstract in a framework at the interface between statistical physics and computational complexity we discuss the extension of the pfaffian formalism for the evaluation of the ising partition function or of the weighted matching polynomial over planar lattices to the general case of non planar graphs the combinatorial features of the method acquire a simple topological character independent of the local details of the lattices as a by product we also give a simple formula for the evaluation of the permanent of 0 1 matrices in terms of determinants

critical ising on the square lattice mixes in polynomial time the ising model is widely regarded as the most studied model of spin systems in statistical physics the focus of this paper is its dynamic stochastic version the glauber dynamics introduced in 1963 and by now the most popular means of sampling the ising measure intensive study throughout the last three decades has yielded a rigorous understanding of the spectral gap of the dynamics on mathbb z 2 everywhere except at criticality while the critical behavior of the ising model has long been the focus for physicists mathematicians have only recently developed an understanding of its critical geometry with the advent of sle cle and new tools to study conformally invariant systems

total variation discrepancy of deterministic random walks for ergodic markov chains abstract motivated by a derandomization of markov chain monte carlo mcmc this paper investigates a deterministic random walk which is a deterministic process analogous to a random walk there is some recent progress in the analysis of the vertex wise discrepancy i e l discrepancy while little is known about the total variation discrepancy i e l 1 discrepancy which plays an important role in the analysis of an fpras based on mcmc this paper investigates the l 1 discrepancy between the expected number of tokens in a markov chain and the number of tokens in its corresponding deterministic random walk first we give a simple but nontrivial upper bound o m t of the l 1 discrepancy for any ergodic markov chains where m is the number of edges of the transition diagram and t is the mixing time of the markov chain then we give a better upper bound o m t for non oblivious deterministic random walks if the corresponding markov chain is ergodic and lazy we also present some lower bounds

discriminative learning of max sum classifiers the max sum classifier predicts n tuple of labels from n tuple of observable variables by maximizing a sum of quality functions defined over neighbouring pairs of labels and observable variables predicting labels as map assignments of a random markov field is a particular example of the max sum classifier learning parameters of the max sum classifier is a challenging problem because even computing the response of such classifier is np complete in general estimating parameters using the maximum likelihood approach is feasible only for a subclass of max sum classifiers with an acyclic structure of neighbouring pairs recently the discriminative methods represented by the perceptron and the support vector machines originally designed for binary linear classifiers have been extended for learning some subclasses of the max sum classifier besides the max sum classifiers with the acyclic neighbouring structure it has been shown that the discriminative learning is possible even with arbitrary neighbouring structure provided the quality functions fulfill some additional constraints in this article we extend the discriminative approach to other three classes of max sum classifiers with an arbitrary neighbourhood structure we derive learning algorithms for two subclasses of max sum classifiers whose response can be computed in polynomial time i the max sum classifiers with supermodular quality functions and ii the max sum classifiers whose response can be computed exactly by a linear programming relaxation moreover we show that the learning problem can be approximately solved even for a general max sum classifier

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

stein s method for concentration inequalities we introduce a version of stein s method for proving concentration and moment inequalities in problems with dependence simple illustrative examples from combinatorics physics and mathematical statistics are provided

on the hardness of sampling independent sets beyond the tree threshold we consider local markov chain monte carlo algorithms for sampling from the weighted distribution of independent sets with activity where the weight of an independent set i is i a recent result has established that gibbs sampling is rapidly mixing in sampling the distribution for graphs of maximum degree d and c it is np hard to approximate the above weighted sum over independent sets to within a factor polynomial in the size of the graph

computing hermite and smith normal forms of triangular integer matrices abstract this paper considers the problem of transforming a triangular integer input matrix to canonical hermite and smith normal form we provide algorithms and prove deterministic running times for both transformation problems that are optimal in the matrix dimension the algorithms are easily implemented assume standard integer arithmetic and admit excellent performance in practice the results presented here lead to a faster algorithm for computing the smith normal form of an arbitrary i e non triangular input matrix

on complexity of the quantum ising model we study complexity of several problems related to the transverse field ising model tim first we consider the problem of estimating the ground state energy known as the local hamiltonian problem lhp it is shown that the lhp for tim on degree 3 graphs is equivalent modulo polynomial reductions to the lhp for general k local stoquastic hamiltonians with any constant k ge 2 this result implies that estimating the ground state energy of tim on degree 3 graphs is a complete problem for the complexity class mathsf stoqma an extension of the classical class mathsf ma as a corollary we complete the complexity classification of 2 local hamiltonians with a fixed set of interactions proposed recently by cubitt and montanaro secondly we study quantum annealing algorithms for finding ground states of classical spin hamiltonians associated with hard optimization problems we prove that the quantum annealing with tim hamiltonians is equivalent modulo polynomial reductions to the quantum annealing with a certain subclass of k local stoquastic hamiltonians this subclass includes all hamiltonians representable as a sum of a k local diagonal hamiltonian and a 2 local stoquastic hamiltonian

sparse reliable graph backbones given a connected graph g and a failure probability p e for each edge e in g the reliability of g is the probability that g remains connected when each edge e is removed independently with probability p e in this paper it is shown that every n vertex graph contains a sparse backbone i e a spanning subgraph with o nlogn edges whose reliability is at least 1 n w 1 times that of g moreover for any pair of vertices s t in g the s t reliability of the backbone namely the probability that s and t remain connected is also at least 1 n w 1 times that of g our proof is based on a polynomial time randomized algorithm for constructing the backbone in addition it is shown that the constructed backbone has nearly the same tutte polynomial as the original graph in the quarter plane x 1 y 1 and hence the graph and its backbone share many additional features encoded by the tutte polynomial