analytical and belief propagation studies of random constraint satisfaction problems with growing domains we study solution space structure and solution finding algorithms of a representative hard random constraint satisfaction problem with growing domains known as model rb using rigorous methods we show that solutions are grouped into disconnected clusters before the theoretical satisfiability phase transition point using the cavity method it is shown that the entropy density obtained by belief propagation bp on random model rb instances which corresponds well to the analytical results vanishes as the control parameter constraint tightness approaches the satisfiability threshold from an algorithmic point of view we find that reinforced bp which performs much better than all existing algorithms allows us to find solutions efficiently for instances in the regime that is very close to the satisfiability transition these results also can shed light on the effectiveness of bp reinforcement on problems with a large number of states

random k sat two moments suffice to cross a sharp threshold many np complete constraint satisfaction problems appear to undergo a phase transition from solubility to insolubility when the constraint density passes through a critical threshold in all such cases it is easy to derive upper bounds on the location of the threshold by showing that above a certain density the first moment expectation of the number of solutions tends to zero we show that in the case of certain symmetric constraints considering the second moment of the number of solutions yields nearly matching lower bounds for the location of the threshold specifically we prove that the threshold for both random hypergraph 2 colorability property b and random not all equal k sat is 2 k 1 ln 2 o 1 as a corollary we establish that the threshold for random k sat is of order theta 2 k resolving a long standing open problem

rigorous inequalities between length and time scales in glassy systems glassy systems are characterized by an extremely sluggish dynamics without any simple sign of long range order it is a debated question whether a correct description of such phenomenon requires the emergence of a large correlation length we prove rigorous bounds between length and time scales implying the growth of a properly defined length when the relaxation time increases our results are valid in a rather general setting which covers finite dimensional and mean field systems as an illustration we discuss the glauber heat bath dynamics of p spin glass models on random regular graphs we present the first proof that a model of this type undergoes a purely dynamical phase transition not accompanied by any thermodynamic singularity

social balance as a satisfiability problem of computer science reduction of frustration was the driving force in an approach to social balance as it was recently considered by antal t antal p l krapivsky and s redner phys rev e 72 036121 2005 we generalize their triad dynamics to k cycle dynamics for arbitrary integer k we derive the phase structure determine the stationary solutions and calculate the time it takes to reach a frozen state the main difference in the phase structure as a function of k is related to k being even or odd as a second generalization we dilute the all to all coupling as considered by antal to a random network with connection probability w 1 interestingly this model can be mapped to a satisfiability problem of computer science the phase of social balance in our original interpretation then becomes the phase of satisfaction of all logical clauses in the satisfiability problem in common to the cases we study the ideal solution without any frustration always exists but the question actually is as to whether this solution can be found by means of a local stochastic algorithm within a finite time the answer depends on the choice of parameters after establishing the mapping between the two classes of models we generalize the social balance problem to a diluted network topology for which the satisfiability problem is usually studied on the other hand in connection with the satisfiability problem we generalize the random local algorithm to a p random local algorithm including a parameter p that corresponds to the propensity parameter in the social balance problem the qualitative effect of the inclusion of this parameter is a bias towards the optimal solution and a reduction of the needed simulation time

the generalized distributive law we discuss a general message passing algorithm which we call the generalized distributive law gdl the gdl is a synthesis of the work of many authors in information theory digital communications signal processing statistics and artificial intelligence it includes as special cases the baum welch algorithm the fast fourier transform fft on any finite abelian group the gallager tanner wiberg decoding algorithm viterbi s algorithm the bcjr algorithm pearl s belief propagation algorithm the shafer shenoy probability propagation algorithm and the turbo decoding algorithm although this algorithm is guaranteed to give exact answers only in certain cases the junction tree condition unfortunately not including the cases of gtw with cycles or turbo decoding there is much experimental evidence and a few theorems suggesting that it often works approximately even when it is not supposed to

t 0 mean field population dynamics approach for the random 3 satisfiability problem during the past decade phase transition phenomena in the random 3 satisfiability 3 sat problem has been intensively studied by statistical physics methods in this work we study the random 3 sat problem by the mean field first step replica symmetry broken cavity theory at the limit of temperature t 0 the reweighting parameter y of the cavity theory is allowed to approach infinity together with the inverse temperature beta with fixed ratio r y beta focusing on the system s space of satisfiable configurations we carry out extensive population dynamics simulations using the technique of importance sampling and we obtain the entropy density s r and complexity sigma r of zero energy clusters at different r values we demonstrate that the population dynamics may reach different fixed points with different types of initial conditions by knowing the trends of s r and sigma r with r we can judge whether a certain type of initial condition is appropriate at a given r value this work complements and confirms the results of several other very recent theoretical studies

optimal stopping of markov processes hilbert space theory approximation algorithms and an application to pricing high dimensional financial derivatives the authors develop a theory characterizing optimal stopping times for discrete time ergodic markov processes with discounted rewards the theory differs from prior work by its view of per stage and terminal reward functions as elements of a certain hilbert space in addition to a streamlined analysis establishing existence and uniqueness of a solution to bellman s equation this approach provides an elegant framework for the study of approximate solutions in particular the authors propose a stochastic approximation algorithm that tunes weights of a linear combination of basis functions in order to approximate a value function they prove that this algorithm converges almost surely and that the limit of convergence has some desirable properties the utility of the approximation method is illustrated via a computational case study involving the pricing of a path dependent financial derivative security that gives rise to an optimal stopping problem with a 100 dimensional state space

superspaces and supersymmetries a theory of graded banach modules over a banach grassmann algebra is developed and applied to differential geometry of super manifolds the explicit structure of superspaces carrying poincare supersymmetry and extended supersymmetry including central charges is described

tricritical points in random combinatorics the sat case the satisfiability sat problem interpolates between different classes of complexity theory and is thought to be of basic interest in understanding the onset of typical case complexity in random combinatorics in this paper a tricritical point in the phase diagram of the random sat problem is analytically computed using the replica approach and found to lie in the range these bounds on are in agreement with previous numerical simulations and rigorous results

communities of solutions in single solution clusters of a random k satisfiability formula the solution space of a k satisfiability k sat formula is a collection of solution clusters each of which contains all the solutions that are mutually reachable through a sequence of single spin flips knowledge of the statistical property of solution clusters is valuable for a complete understanding of the solution space structure and the computational complexity of the random k sat problem this paper explores single solution clusters of random 3 and 4 sat formulas through unbiased and biased random walk processes and the replica symmetric cavity method of statistical physics we find that the giant connected component of the solution space has already formed many different communities when the constraint density of the formula is still lower than the solution space clustering transition point solutions of the same community are more similar with each other and more densely connected with each other than with the other solutions the entropy density of a solution community is calculated using belief propagation and is found to be different for different communities of the same cluster when the constraint density is beyond the clustering transition point the same behavior is observed for the solution clusters reached by several stochastic search algorithms taking together the results of this work suggest a refined picture on the evolution of the solution space structure of the random k sat problem they may also be helpful for designing heuristic algorithms

solving satisfiability problems by fluctuations the dynamics of stochastic local search algorithms stochastic local search algorithms are frequently used to numerically solve hard combinatorial optimization or decision problems we give numerical and approximate analytical descriptions of the dynamics of such algorithms applied to random satisfiability problems we find two different dynamical regimes depending on the number of constraints per variable for low constraintness the problems are solved efficiently i e in linear time for higher constraintness the solution times become exponential we observe that the dynamical behavior is characterized by a fast equilibration and fluctuations around this equilibrium if the algorithm runs long enough an exponentially rare fluctuation towards a solution appears

dynamic phase transition for decoding algorithms the state of the art error correcting codes are based on large random constructions random graphs random permutations etc and are decoded by linear time iterative algorithms because of these features they are remarkable examples of diluted mean field spin glasses both from the static and dynamic points of view we analyze the behavior of decoding algorithms by mapping them onto statistical physics models this allows us to understand the intrinsic i e algorithm independent features of this behavior

when a local hamiltonian must be frustration free a broad range of quantum optimization problems can be phrased as the question of whether a specific system has a ground state at zero energy i e whether its hamiltonian is frustration free frustration free hamiltonians in turn play a central role for constructing and understanding new phases of matter in quantum many body physics unfortunately determining whether this is the case is known to be a complexity theoretically intractable problem this makes it highly desirable to search for efficient heuristics and algorithms to at least partially answer this question here we prove a general criterion a sufficient condition under which a local hamiltonian is guaranteed to be frustration free by lifting shearer s theorem from classical probability theory to the quantum world remarkably evaluating this condition proceeds via a fully classical analysis of a hardcore lattice gas at negative fugacity on the hamiltonian s interaction graph which as a statistical mechanics problem is of interest in its own right we concretely apply this criterion to local hamiltonians on various regular lattices while bringing to bear the tools of spin glass physics that permit us to obtain new bounds on the satisfiable to unsatisfiable transition in random quantum satisfiability we are then led to natural conjectures for when such bounds will be tight as well as to a novel notion of universality for these computer science problems besides providing concrete algorithms leading to detailed and quantitative insights this work underscores the power of marrying classical statistical mechanics with quantum computation and complexity theory

efficient computation of hedging parameters for discretely exercisable options we propose an algorithm to calculate confidence intervals for the values of hedging parameters of discretely exercisable options using monte carlo simulation the algorithm is based on a combination of the duality formulation of the optimal stopping problem for pricing discretely exercisable options and monte carlo estimation of hedging parameters for european options we show that the width of the confidence interval for a hedging parameter decreases with an increase in the computer budget asymptotically at the same rate as the width of the confidence interval for the price of the option the method can handle arbitrary payoff functions general diffusion processes and a large number of random factors we also present a fast heuristic alternative method and use our method to evaluate its accuracy

spanning trees in random satisfiability problems working with tree graphs is always easier than with loopy ones and spanning trees are the closest tree like structures to a given graph we find a correspondence between the solutions of random k satisfiability problem and those of spanning trees in the associated factor graph we introduce a modified survey propagation algorithm which returns null edges of the factor graph and helps us to find satisfiable spanning trees this allows us to study organization of satisfiable spanning trees in the space spanned by spanning trees

sequential and cooperative sensing for multi channel cognitive radios effective spectrum sensing is a critical prerequisite for multi channel cognitive radio cr networks where multiple spectrum bands are sensed to identify transmission opportunities while preventing interference to the primary users the present paper develops sequential spectrum sensing algorithms which explicitly take into account the sensing time overhead and optimize a performance metric capturing the effective average data rate of cr transmitters a constrained dynamic programming problem is formulated to obtain the policy that chooses the best time to stop taking measurements and the best set of channels to access for data transmission while adhering to hard collision constraints imposed to protect primary links given the associated lagrange multipliers the optimal access policy is obtained in closed form and the subsequent problem reduces to an optimal stopping problem a basis expansion based sub optimal strategy is employed to mitigate the prohibitive computational complexity of the optimal stopping policy a novel on line implementation based on the recursive least squares rls algorithm along with a stochastic dual update procedure is then developed to obviate the lengthy training phase of the batch scheme cooperative sequential sensing generalizations are also provided with either raw or quantized measurements collected at a central processing unit the numerical results presented verify the efficacy of the proposed algorithms

multivariate convex regression with adaptive partitioning we propose a new nonparametric method for multivariate regression subject to convexity or concavity constraints on the response function convexity constraints are common in economics statistics operations research financial engineering and optimization but there is currently no multivariate method that is stable and computationally feasible for more than a few thousand observations we introduce convex adaptive partitioning cap which creates a globally convex regression model from locally linear estimates fit on adaptively selected covariate partitions cap is a computationally efficient consistent method for convex regression we demonstrate empirical performance by comparing the performance of cap to other shape constrained and unconstrained regression methods for predicting weekly wages and value function approximation for pricing american basket options

realizability of point processes there are various situations in which it is natural to ask whether a given collection of k functions j r 1 r j j 1 k defined on a set x are the first k correlation functions of a point process on x here we describe some necessary and sufficient conditions on the j s for this to be true our primary examples are x d x d and x an arbitrary finite set in particular we extend a result by ambartzumian and sukiasian showing realizability at sufficiently small densities 1 r typically if any realizing process exists there will be many even an uncountable number in this case we prove when x is a finite set the existence of a realizing gibbs measure with k body potentials which maximizes the entropy among all realizing measures we also investigate in detail a simple example in which a uniform density and translation invariant 2 are specified on there is a gap between our best upper bound on possible values of and the largest for which realizability can be established

discrete supergroups and super riemann surfaces the properties of discrete supergroups that represent the homotopy groups of super riemann surfaces are emphasized after the study of the superforms invariant with respect to these supergroups the set of superconformal differentials are defined and an example is given of those by defining poincare s super theta functions some remarks on a possible definition of the jacobi variety of a super riemann surface are given

q learning and enhanced policy iteration in discounted dynamic programming we consider the classical finite state discounted markovian decision problem and we introduce a new policy iteration like algorithm for finding the optimal state costs or q factors the main difference is in the policy evaluation phase instead of solving a linear system of equations our algorithm requires solving an optimal stopping problem the solution of this problem may be inexact with a finite number of value iterations in the spirit of modified policy iteration the stopping problem structure is incorporated into the standard q learning algorithm to obtain a new method that is intermediate between policy iteration and q learning value iteration thanks to its special contraction properties our method overcomes some of the traditional convergence difficulties of modified policy iteration and admits asynchronous deterministic and stochastic iterative implementations with lower overhead and or more reliable convergence over existing q learning schemes furthermore for large scale problems where linear basis function approximations and simulation based temporal difference implementations are used our algorithm addresses effectively the inherent difficulties of approximate policy iteration due to inadequate exploration of the state and control spaces

on the satisfiability threshold and clustering of solutions of random 3 sat formulas we study the structure of satisfying assignments of a random 3 sat formula in particular we show that a random formula of density a 4 453 almost surely has no non trivial core assignments core assignments are certain partial assignments that can be extended to satisfying assignments and have been studied recently in connection with the survey propagation heuristic for random sat their existence implies the presence of clusters of solutions and they have been shown to exist with high probability below the satisfiability threshold for k sat with k 9 d achlioptas f ricci tersenghi on the solution space geometry of random constraint satisfaction problems in proc 38th acm symp theory of computing stoc 2006 pp 130 139 our result implies that either this does not hold for 3 sat or the threshold density for satisfiability in 3 sat lies below 4 453 the main technical tool that we use is a novel simple application of the first moment method

constructing free energy approximations and generalized belief propagation algorithms important inference problems in statistical physics computer vision error correcting coding theory and artificial intelligence can all be reformulated as the computation of marginal probabilities on factor graphs the belief propagation bp algorithm is an efficient way to solve these problems that is exact when the factor graph is a tree but only approximate when the factor graph has cycles we show that bp fixed points correspond to the stationary points of the bethe approximation of the free energy for a factor graph we explain how to obtain region based free energy approximations that improve the bethe approximation and corresponding generalized belief propagation gbp algorithms we emphasize the conditions a free energy approximation must satisfy in order to be a valid or maxent normal approximation we describe the relationship between four different methods that can be used to generate valid approximations the bethe method the junction graph method the cluster variation method and the region graph method finally we explain how to tell whether a region based approximation and its corresponding gbp algorithm is likely to be accurate and describe empirical results showing that gbp can significantly outperform bp

pathwise optimization for optimal stopping problems we introduce the pathwise optimization po method a new convex optimization procedure to produce upper and lower bounds on the optimal value the price of a high dimensional optimal stopping problem the po method builds on a dual characterization of optimal stopping problems as optimization problems over the space of martingales which we dub the martingale duality approach we demonstrate via numerical experiments that the po method produces upper bounds of a quality comparable with state of the art approaches but in a fraction of the time required for those approaches as a by product it yields lower bounds and suboptimal exercise policies that are substantially superior to those produced by state of the art methods the po method thus constitutes a practical and desirable approach to high dimensional pricing problems furthermore we develop an approximation theory relevant to martingale duality approaches in general and the po method in particular our analysis provides a guarantee on the quality of upper bounds resulting from these approaches and identifies three key determinants of their performance the quality of an input value function approximation the square root of the effective time horizon of the problem and a certain spectral measure of predictability of the underlying markov chain as a corollary to this analysis we develop approximation guarantees specific to the po method finally we view the po method and several approximate dynamic programming methods for high dimensional pricing problems through a common lens and in doing so show that the po method dominates those alternatives this paper was accepted by wei xiong stochastic models and simulation

long range anticorrelations and non gaussian behavior of the heartbeat we find that the successive increments in the cardiac beat to beat interals of healthy subjects display scale invariant long range anticorrelations up to 10 4 heart beats furthermore we find that the histogram for the heartbeat intervals increments is well described by a l evy stable distribution for a group of subjects with severe heart disease we find that the distribution is unchanged but the long range correlations vanish therefore the different scaling behavior in health and disease must relate to the underlying dynamics of the heartbeat

variational calculus on supermanifolds and invariance properties of superspace field theories the foundations of a variational calculus on fibered supermanifolds are outlined giving applications to the formulation of superspace field theories utiyama theorems and conservation laws related to local gauge and general invariance of the theory are proved

random subcubes as a toy model for constraint satisfaction problems we present an exactly solvable random subcube model inspired by the structure of hard constraint satisfaction and optimization problems our model reproduces the structure of the solution space of the random k satisfiability and k coloring problems and undergoes the same phase transitions as these problems the comparison becomes quantitative in the large k limit distance properties as well the x satisfiability threshold are studied the model is also generalized to define a continuous energy landscape useful for studying several aspects of glassy dynamics

is sp bp the survey propagation sp algorithm for solving k sat problems has been shown recently as an instance of the belief propagation bp algorithm in this paper we show that for general constraint satisfaction problems sp may not be reducible from bp we also establish the conditions under which such a reduction is possible along our development we present a unification of the existing sp algorithms in terms of a probabilistically interpretable iterative procedure weighted probabilistic token passing

projected equation methods for approximate solution of large linear systems we consider linear systems of equations and solution approximations derived by projection on a low dimensional subspace we propose stochastic iterative algorithms based on simulation which converge to the approximate solution and are suitable for very large dimensional problems the algorithms are extensions of recent approximate dynamic programming methods known as temporal difference methods which solve a projected form of bellman s equation by using simulation based approximations to this equation or by using a projected value iteration method

an approximate moving boundary method for american option pricing we present a method to solve the free boundary problem that arises in the pricing of classical american options such free boundary problems arise when one attempts to solve optimal stopping problems set in continuous time american option pricing is one of the most popular optimal stopping problems considered in literature the method presented in this paper primarily shows how one can leverage on a one factor approximation and the moving boundary approach to construct a solution mechanism the result is an algorithm that has superior runtimes accuracy balance to other computational methods that are available to solve the free boundary problems exhaustive comparisons to other pricing methods are provided we also discuss a variant of the proposed algorithm that allows for the computation of only one option price rather than the entire price function when the requirement is such

two solutions to diluted p spin models and xorsat problems we derive analytical solutions for p spin models with finite connectivity at zero temperature these models are the statistical mechanics equivalent of p xorsat problems in theoretical computer science we give a full characterization of the phase diagram location of the phase transitions static and dynamic together with a description of the clustering phenomenon taking place in configurational space we use two alternative methods the cavity approach and a rigorous derivation

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

behavior of heuristics on large and hard satisfiability problems we study the behavior of a heuristic for solving random satisfiability problems by stochastic local search near the satisfiability threshold the heuristic for average satisfiability asat is si

convergence results for some temporal difference methods based on least squares we consider finite state markov decision processes and prove convergence and rate of convergence results for certain least squares policy evaluation algorithms of the type known as lspe lambda these are temporal difference methods for constructing a linear function approximation of the cost function of a stationary policy within the context of infinite horizon discounted and average cost dynamic programming we introduce an average cost method patterned after the known discounted cost method and we prove its convergence for a range of constant stepsize choices we also show that the convergence rate of both the discounted and the average cost methods is optimal within the class of temporal difference methods analysis and experiment indicate that our methods are substantially and often dramatically faster than td lambda as well as more reliable

option pricing under a double exponential jump diffusion model this paper aims to extend the analytical tractability of the black scholes model to alternative models with arbitrary jump size distributions more precisely we propose a jump diffusion model for asset prices whose jump sizes have a mixed exponential distribution which is a weighted average of exponential distributions but with possibly negative weights the new model extends existing models such as hyperexponential and double exponential jump diffusion models because the mixed exponential distribution can approximate any distribution as closely as possible including the normal distribution and various heavy tailed distributions the mixed exponential jump diffusion model can lead to analytical solutions for laplace transforms of prices and sensitivity parameters for path dependent options such as lookback and barrier options the laplace transforms can be inverted via the euler inversion algorithm numerical experiments indicate that the formulae are easy to implement and accurate the analytical solutions are made possible mainly because we solve a high order integro differential equation explicitly a calibration example for spy options shows that the model can provide a reasonable fit even for options with very short maturity such as one day this paper was accepted by michael fu stochastic models and simulation

quantum max flow min cut the classical max flow min cut theorem describes transport through certain idealized classical networks we consider the quantum analog for tensor networks by associating an integral capacity to each edge and a tensor to each vertex in a flow network we can also interpret it as a tensor network and more specifically as a linear map from the input space to the output space the quantum max flow is defined to be the maximal rank of this linear map over all choices of tensors the quantum min cut is defined to be the minimum product of the capacities of edges over all cuts of the tensor network we show that unlike the classical case the quantum max flow min cut conjecture is not true in general under certain conditions e g when the capacity on each edge is some power of a fixed integer the quantum max flow is proved to equal the quantum min cut however concrete examples are also provided where the equality does not hold we also found connections of quantum max flow min cut with entropy of entanglement and the quantum satisfiability problem we speculate that the phenomena revealed may be of interest both in spin systems in condensed matter and in quantum gravity

possible sets of autocorrelations and the simplex algorithm the problem of imposing a set of correlations of any order on binary sequences is addressed the entropy of infinitely long sequences obeying such a given set was calculated in previous works using the saddle point method and it was observed that a finite fraction of sets are characterized by a non extensive entropy in this paper the region of finite entropy the allowed region of sets of correlations is found to be a convex hyper polygon in the space of correlation sets using the simplex algorithm outside of this region the simplex solution indicates that sequences obeying the correlations cannot be found therefore the entropy is in particular the boundaries of the allowed region for c1 cm are presented at the boundaries the entropy drops in a first order phase transition fashion and this drop can be explained from a combinatorial point of view finally we observe that the fraction of the volume occupied by allowed correlation sets drops exponentially with the number of correlations imposed and a qualitative explanation of this scaling phenomenon is provided

maximally flexible solutions of a random k satisfiability formula random k satisfiability k sat is a paradigmatic model system for studying phase transitions in constraint satisfaction problems and for developing empirical algorithms the statistical properties of the random k sat solution space have been extensively investigated but most earlier efforts focused on solutions that are typical here we consider maximally flexible solutions which satisfy all the constraints only using the minimum number of variables such atypical solutions have high internal entropy because they contain a maximum number of null variables which are completely free to choose their states each maximally flexible solution indicates a dense region of the solution space we estimate the maximum fraction of null variables by the replica symmetric cavity method and implement message passing algorithms to construct maximally flexible solutions for single k sat instances

complexity of stoquastic frustration free hamiltonians we study several problems related to properties of nonnegative matrices that arise at the boundary between quantum and classical probabilistic computation our results are twofold first we identify a large class of quantum hamiltonians describing systems of qubits for which the adiabatic evolution can be efficiently simulated on a classical probabilistic computer these are stoquastic local hamiltonians with a frustration free ground state a hamiltonian belongs to this class iff it can be represented as h sum a h a where 1 every term h a acts nontrivially on a constant number of qubits 2 every term h a has real nonpositive off diagonal matrix elements in the standard basis and 3 the ground state of h is a ground state of every term h a second we generalize the cook levin theorem proving np completeness of the satisfiability problem to the complexity class ma merlin arthur games a probabilistic analogue of np specifically we construct a quantum version of the k sat problem which we call stoquastic k sat such that stoquastic k sat is contained in ma for any constant k and any promise problem in ma is karp reducible to stoquastic 6 sat this result provides the first nontrivial example of a ma complete promise problem

error bounds for approximations from projected linear equations we consider linear fixed point equations and their approximations by projection on a low dimensional subspace we derive new bounds on the approximation error of the solution which are expressed in terms of low dimensional matrices and can be computed by simulation when the fixed point mapping is a contraction as is typically the case in markov decision processes mdp one of our bounds is always sharper than the standard contraction based bounds and another one is often sharper the former bound is also tight in a worst case sense our bounds also apply to the noncontraction case including policy evaluation in mdp with nonstandard projections that enhance exploration there are no error bounds currently available for this case to our knowledge

detecting the solution space of vertex cover by mutual determinations and backbones to solve the combinatorial optimization problems especially the minimal vertex cover problem with high efficiency is a significant task in theoretical computer science and many other subjects aiming at detecting the solution space of vertex cover a new structure named mutual determination is defined and discovered for vertex cover on general graphs which results in the emergence of strong correlations among the unfrozen nodes based on the backbones and mutual determinations with node ranks by leaf removal we propose a mutual determination and backbone evolution algorithm to achieve the reduced solution graph which provides a graphical expression of the solution space of vertex cover by this algorithm the whole solution space and detailed structures such as backbones can be obtained strictly when there is no leaf removal core on the given graph compared with the current algorithms the mutual determination and backbone evolution algorithm performs as well as the replica symmetry one in a certain interval but has a small gap higher than the replica symmetric breaking one and has a relatively small error for the exact results the algorithm with the mutual determination provides a new viewpoint to solve vertex cover and understand the organizations of the solution spaces and the reduced solution graph gives an alternative way to catch detailed information of the ground steady states

gibbs states and the set of solutions of random constraint satisfaction problems abstract an instance of a random constraint satisfaction problem defines a random subset the set of solutions of a large product space x n the set of assignments we consider two prototypical problem ensembles random k satisfiability and q coloring of random regular graphs and study the uniform measure with support on s as the number of constraints per variable increases this measure first decomposes into an exponential number of pure states clusters and subsequently condensates over the largest such states above the condensation point the mass carried by the n largest states follows a poisson dirichlet process for typical large instances the two transitions are sharp we determine their precise location further we provide a formal definition of each phase transition in terms of different notions of correlation between distinct variables in the problem the degree of correlation naturally affects the performances of many search sampling algorithms empirical evidence suggests that local monte carlo markov chain strategies are effective up to the clustering phase transition and belief propagation up to the condensation point finally refined message passing techniques such as survey propagation may also beat this threshold message passing algorithms phase transitions random graphs

low autocorrelated multiphase sequences the interplay between the ground state energy of the generalized bernasconi model to multiphase and the minimal value of the maximal autocorrelation function c m a x max k c k k 1 n i is examined analytically in the thermodynamic limit where the main results are a for the binary case the minimal value of c m a x over all sequences of length n minc m a x is 0 435 n significantly smaller than the typical value for random sequences o log n n b a new method to approximate f m a x is obtained using the observation of data collapse c minc m a x is obtained in an energy which is about 30 above the ground state energy of the generalized bernasconi model independent of the number of phases m d for a given m minc m a x n m indicating that for m n minc m a x 1 i e a generalized barker code exists the analytical results are confirmed by simulations

clustering of solutions in the random satisfiability problem using elementary rigorous methods we prove the existence of a clustered phase in the random k sat problem for k geq 8 in this phase the solutions are grouped into clusters which are far away from each other the results are in agreement with previous predictions of the cavity method and give a rigorous confirmation to one of its main building blocks it can be generalized to other systems of both physical and computational interest

phase transitions in the q coloring of random hypergraphs we study in this paper the structure of solutions in the random hypergraph coloring problem and the phase transitions they undergo when the density of constraints is varied hypergraph coloring is a constraint satisfaction problem where each constraint includes k variables that must be assigned one out of q colors in such a way that there are no monochromatic constraints i e there are at least two distinct colors in the set of variables belonging to every constraint this problem generalizes naturally coloring of random graphs k 2 and bicoloring of random hypergraphs q 2 both of which were extensively studied in past works the study of random hypergraph coloring gives us access to a case where both the size q of the domain of the variables and the arity k of the constraints can be varied at will our work provides explicit values and predictions for a number of phase transitions that were discovered in other constraint satisfaction problems but never evaluated before in hypergraph coloring among other cases we revisit the hypergraph bicoloring problem q 2 where we find that for k 3 and k 4 the colorability threshold is not given by the one step replica symmetry breaking analysis as the latter is unstable towards more levels of replica symmetry breaking we also unveil and discuss the coexistence of two different 1rsb solutions in the case of q 2 k 4 finally we present asymptotic expansions for the density of constraints at which various phase transitions occur in the limit where q and or k diverge

on n 4 super riemann surfaces and the superconformal semigroup the finite n 4superconformal scf transformations areobtained and classified into three types by means of the permanent of specific matrices the properties of the latter are studied in connection with complex plane non euclidean geometry the explicit expression for the n 4 scf transformations allows them to be used as the transition functions for n 4 split and non split super riemann surfaces and scf embeddings the existence of the semigroup of scf transformations is indicated

on the semigroup nature of superconformal symmetry a semigroup of n 1 superconformal transformations is introduced and analyzed noninvertible ones can describe transitions from body to soul and form a proper ideal containing a set of nilpotent transformations the projective superspace is also considered transformations twisting the parity of a tangent space are brought in they can be a nonsuperconformal square root of noninvertible superconformal transformations and the analogs of the poincare metric and conformal invariance are suggested for them

a new look at survey propagation and its generalizations this article provides a new conceptual perspective on survey propagation which is an iterative algorithm recently introduced by the statistical physics community that is very effective in solving random k sat problems even with densities close to the satisfiability threshold we first describe how any sat formula can be associated with a novel family of markov random fields mrfs parameterized by a real number 0 1 we then show that applying belief propagation a well known message passing technique for estimating marginal probabilities to this family of mrfs recovers a known family of algorithms ranging from pure survey propagation at one extreme e 1 to standard belief propagation on the uniform distribution over sat assignments at the other extreme e 0 configurations in these mrfs have a natural interpretation as partial satisfiability assignments on which a partial order can be defined we isolate cores as minimal elements in this partial ordering which are also fixed points of survey propagation and the only assignments with positive probability in the mrf for e 1 our experimental results for k e 3 suggest that solutions of random formulas typically do not possess non trivial cores this makes it necessary to study the structure of the space of partial assignments for 0 can be viewed as a smoothed version of the uniform distribution over satisfying assignments e 0 finally we isolate properties of gibbs sampling and message passing algorithms that are typical for an ensemble of k sat problems

a quantum lovasz local lemma the lovasz local lemma lll is a powerful tool in probability theory to show the existence of combinatorial objects meeting a prescribed collection of weakly dependent criteria we show that the lll extends to a much more general geometric setting where events are replaced with subspaces and probability is replaced with relative dimension which allows to lower bound the dimension of the intersection of vector spaces under certain independence conditions our result immediately applies to the k qsat problem quantum analog of k sat for instance we show that any collection of rank 1 projectors with the property that each qubit appears in at most 2k e k of them has a joint satisfiable state we then apply our results to the recently studied model of random k qsat recent works have shown that the satisfiable region extends up to a density of 1 in the large k limit where the density is the ratio of projectors to qubits using a hybrid approach building on work by laumann et al 2009 2010 we greatly extend the known satisfiable region for random k qsat to a density of 2k k2 since our tool allows us to show the existence of joint satisfying states without the need to construct them we are able to penetrate into regions where the satisfying states are conjectured to be entangled avoiding the need to construct them which has limited previous approaches to product states

using convex switching techniques for partially observable decision processes we present and examine a novel method for obtaining solutions to specific discrete time optimal control problems our approach is based on linear state dynamics and convexity assumptions commonly satisfied in practical applications we show that the important class of optimal switching problems under partial observation is covered by our methodology and we exploit specific model features to achieve simple algorithmic form of a numerical solution

temporal difference methods for general projected equations we consider projected equations for approximate solution of high dimensional fixed point problems within low dimensional subspaces we introduce an analytical framework based on an equivalence with variational inequalities and algorithms that may be implemented with low dimensional simulation these algorithms originated in approximate dynamic programming dp where they are collectively known as temporal difference td methods even when specialized to dp our methods include extensions new versions of td methods which offer special implementation advantages and reduced overhead over the standard lstd and lspe methods and can deal with near singularity in the associated matrix inversion we develop deterministic iterative methods and their simulation based versions and we discuss a sharp qualitative distinction between them the performance of the former is greatly affected by direction and feature scaling yet the latter have the same asymptotic convergence rate regardless of scaling because of their common simulation induced performance bottleneck

speedup learning for repair based search by identifying redundant steps repair based search algorithms start with an initial solution and attempt to improve it by iteratively applying repair operators such algorithms can often handle large scale problems that may be difficult for systematic search algorithms nevertheless the computational cost of solving such problems is still very high we observed that many of the repair steps applied by such algorithms are redundant in the sense that they do not eventually contribute to finding a solution such redundant steps are particularly harmful in repair based search where each step carries high cost due to the very high branching factor typically associated with it accurately identifying and avoiding such redundant steps would result in faster local search without harming the algorithm s problem solving ability in this paper we propose a speedup learning methodology for attaining this goal it consists of the following steps defining the concept of a redundant step acquiring this concept during off line learning by analyzing solution paths for training problems tagging all the steps along the paths according to the redundancy definition and using an induction algorithm to infer a classifier based on the tagged examples and using the acquired classifier to filter out redundant steps while solving unseen problems our algorithm was empirically tested on instances of real world employee timetabling problems etp the problem solver to be improved is based on one of the best methods for solving some large etp instances our results show a significant improvement in speed for test problems that are similar to the given example problems

searching stochastically generated multi abstraction level design spaces abstract we present a new algorithm called highest utility first search hufs for searching trees characterized by a large branching factor the absence of a heuristic to compare nodes at different levels of the tree and a child generator that is both expensive to run and stochastic in nature such trees arise naturally for instance in problems which involve candidate designs at several levels of abstraction and which use stochastic optimizers such as genetic algorithms or simulated annealing to generate a candidate at one level from a parent at the previous level hufs is applicable when there is a class of related problems from which many specific problems will need to be solved this paper explains the hufs algorithm and presents experimental results comparing hufs with alternative methods

an analysis of temporal difference learning with function approximation we discuss the temporal difference learning algorithm as applied to approximating the cost to go function of an infinite horizon discounted markov chain the algorithm we analyze updates parameters of a linear function approximator online during a single endless trajectory of an irreducible aperiodic markov chain with a finite or infinite state space we present a proof of convergence with probability one a characterization of the limit of convergence and a bound on the resulting approximation error furthermore our analysis is based on a new line of reasoning that provides new intuition about the dynamics of temporal difference learning in addition to proving new and stronger positive results than those previously available we identify the significance of online updating and potential hazards associated with the use of nonlinear function approximators first we prove that divergence may occur when updates are not based on trajectories of the markov chain this fact reconciles positive and negative results that have been discussed in the literature regarding the soundness of temporal difference learning second we present an example illustrating the possibility of divergence when temporal difference learning is used in the presence of a nonlinear function approximator

sharp thresholds of graph properties and the sat problem consider g n p to be the probability space of random graphs on n vertices with edge probability p we will be considering subsets of this space defined by monotone graph properties a monotone graph property p is a property of graphs such that a p is invariant under graph automorphisims b if graph h has property p then so does any graph g having h as a subgraph a monotone symmetric family of graphs is a family defined by such a property one of the first observations made about random graphs by erdos and renyi in their seminal work on random graph theory 12 was the existence of threshold phenomena the fact that for many interesting properties p the probability of p appearing in g n p exhibits a sharp increase at a certain critical value of the parameter p bollobas and thomason proved the existence of threshold functions for all monotone set properties 6 and in 14 it is shown that this behavior is quite general and that all monotone graph properties exhibit threshold behavior i e the probability of their appearance increases from values very close to 0 to values close to 1 in a very small interval more precise analysis of the size of the threshold interval is done in 7 this threshold behavior which occurs in various settings which arise in combinatorics and computer science is an instance of the phenomenon of phase transitions which is the subject of much interest in statistical physics one of the main questions that arises in studying phase transitions is how sharp is the transition for example one of the motivations for this paper arose from the question of the sharpness of the phase transition for the property of satisfiability of a random kcnf boolean formula nati linial who introduced me to this problem suggested that although much concrete analysis was being performed on this problem the best approach would be to find general conditions for sharpness of the phase transition answering the question posed in 14 as to the relation between the length of the threshold interval and the value of the critical probability in this paper we indeed introduce a simple condition and prove it is sufficient stated roughly in the setting of random graphs the main theorem states that if a property has a coarse threshold then it can be approximated by the property of having certain given graphs as a subgraph this condition can be applied in a more

entropy landscape and non gibbs solutions in constraint satisfaction problems we study the entropy landscape of solutions for the bicoloring problem in random graphs a representative difficult constraint satisfaction problem our goal is to classify which types of clusters of solutions are addressed by different algorithms in the first part of the study we use the cavity method to obtain the number of clusters with a given internal entropy and determine the phase diagram of the problem e g dynamical rigidity and satisfiability unsatisfiability sat unsat transitions in the second part of the paper we analyze different algorithms and locate their behavior in the entropy landscape of the problem for instance we show that a smoothed version of a decimation strategy based on belief propagation is able to find solutions belonging to subdominant clusters even beyond the so called rigidity transition where the thermodynamically relevant clusters become frozen these nonequilibrium solutions belong to the most probable unfrozen clusters

the two parameter poisson dirichlet distribution derived from a stable subordinator the two parameter poisson dirichlet distribution denoted mathsf pd alpha theta is a probability distribution on the set of decreasing positive sequences with sum 1 the usual poisson dirichlet distribution with a single parameter theta introduced by kingman is mathsf pd 0 theta known properties of mathsf pd 0 theta including the markov chain description due to vershik shmidt and ignatov are generalized to the two parameter case the size biased random permutation of mathsf pd alpha theta is a simple residual allocation model proposed by engen in the context of species diversity and rediscovered by perman and the authors in the study of excursions of brownian motion and bessel processes for 0 alpha 1 mathsf pd alpha 0 is the asymptotic distribution of ranked lengths of excursions of a markov chain away from a state whose recurrence time distribution is in the domain of attraction of a stable law of index alpha formulae in this case trace back to work of darling lamperti and wendel in the 1950s and 1960s the distribution of ranked lengths of excursions of a one dimensional brownian motion is mathsf pd 1 2 0 and the corresponding distribution for a brownian bredge is mathsf pd 1 2 1 2 the mathsf pd alpha 0 and mathsf pd alpha alpha distributions admit a similar interpretation in terms of the ranked lengths of excursions of a semistable markov process whose zero set is the range of a stable subordinator of index alpha

on the body of supermanifolds the problem of constructing the body of a g manifold is considered it is shown that any such manifold is foliated and the body is defined to be the space of the leaves of this foliation under certain regularity conditions on the foliation the body is a smooth finite dimensional real manifold

from one solution of a 3 satisfiability formula to a solution cluster frozen variables and entropy a solution to a 3 satisfiability 3 sat formula can be expanded into a cluster all other solutions of which are reachable from this one through a sequence of single spin flips some variables in the solution cluster are frozen to the same spin values by one of two different mechanisms frozen core formation and long range frustrations while frozen cores are identified by a local whitening algorithm long range frustrations are very difficult to trace and they make an entropic belief propagation bp algorithm fail to converge for the bp algorithm to reach a fixed point the spin values of a tiny fraction of variables chosen according to the whitening algorithm are externally fixed during the iteration from the calculated entropy values we infer that for a large random 3 sat formula with constraint density close to the satisfiability threshold the solutions obtained by the survey propagation or walksat algorithms belong neither to the most dominating clusters of the formula nor to the most abundant clusters this work indicates that a single solution cluster of a random 3 sat formula may have further community structures

pairs of sat assignments in random boolean formulae we investigate geometrical properties of the random k satisfiability problem using the notion of x satisfiability a formula is x satisfiable is there exist two sat assignments differing in nx variables we show the existence of a sharp threshold for this property as a function of the clause density for large enough k we prove that there exists a region of clause density below the satisfiability threshold where the landscape of hamming distances between sat assignments experiences a gap pairs of sat assignments exist at small x and around x 12 but they do not exist at intermediate values of x this result is consistent with the clustering scenario which is at the heart of the recent heuristic analysis of satisfiability using statistical physics analysis the cavity method and its algorithmic counterpart the survey propagation algorithm our method uses elementary probabilistic arguments first and second moment methods and might be useful in other problems of computational and physical interest where similar phenomena appear

rigorous low temperature results for the mean field p spins interaction model we prove that just below the critical temperature the mean field p spins interaction model for p suitably large spontaneously decomposes into different states the asymptotic overlaps between any two different states are zero under a mild unproven hypothesis on the weight distribution of these states we prove that they are pure states this situation is called in physics one level of symmetry breaking

clustering phase of a general constraint satisfaction problem model d k csp abstract relation between problem hardness and solution space structure is an important research aspect model d k csp is a random model of constraint satisfaction problem which generates very hard instances when r 1 or r is near 1 where r represents normalized constraint density we study the clustering phase of the model and find that if r is below and close to 1 the solution space contains many widely distributed and well separated small frozen clusters which should be the reason why the generated instances are hard to solve

landscape analysis of constraint satisfaction problems we discuss an analysis of constraint satisfaction problems such as sphere packing k sat and graph coloring in terms of an effective energy landscape several intriguing geometrical properties of the solution space become in this light familiar in terms of the well studied ones of rugged glassy energy landscapes a benchmark algorithm naturally suggested by this construction finds solutions in polynomial time up to a point beyond the clustering and in some cases even the thermodynamic transitions this point has a simple geometric meaning and can be in principle determined with standard statistical mechanical methods thus pushing the analytic bound up to which problems are guaranteed to be easy we illustrate this for the graph 3 and 4 coloring problem for packing problems the present discussion allows to better characterize the j point proposed as a systematic definition of random close packing and to place it in the context of other theories of glasses

supermanifolds and brs transformations we use a supermanifold formalism to support a geometrical interpretation of a recently proposed superfield formulation of brs becchi ronet stora and anti brs transformations

statistical mechanics of the random k satisfiability model the random k satisfiability problem consisting in verifying the existence of an assignment of n boolean variables that satisfy a set of m ensuremath alpha n random logical clauses containing k variables each is studied using the replica symmetric framework of diluted disordered systems we present an exact iterative scheme for the replica symmetric functional order parameter together for the different cases of interest k 2 kg 3 and k ensuremath gg 1 the calculation of the number of solutions which allowed us phys rev lett 76 3881 1996 to predict a first order jump at the threshold where the boolean expressions become unsatisfiable with probability one is thoroughly displayed in the case k 2 the rigorously known critical value ensuremath alpha 1 of the number of clauses per boolean variable is recovered while for kg 3 we show that the system exhibits a replica symmetry breaking transition the annealed approximation is proven to be exact for large k

origin of the computational hardness for learning with binary synapses through supervised learning in a binary perceptron one is able to classify an extensive number of random patterns by a proper assignment of binary synaptic weights however to find such assignments in practice is quite a nontrivial task the relation between the weight space structure and the algorithmic hardness has not yet been fully understood to this end we analytically derive the franz parisi potential for the binary perceptron problem by starting from an equilibrium solution of weights and exploring the weight space structure around it our result reveals the geometrical organization of the weight space the weight space is composed of isolated solutions rather than clusters of exponentially many close by solutions the pointlike clusters far apart from each other in the weight space explain the previously observed glassy behavior of stochastic local search heuristics

phase transitions in the coloring of random graphs we consider the problem of coloring the vertices of a large sparse random graph with a given number of colors so that no adjacent vertices have the same color using the cavity method we present a detailed and systematic analytical study of the space of proper colorings solutions we show that for a fixed number of colors and as the average vertex degree number of constraints increases the set of solutions undergoes several phase transitions similar to those observed in the mean field theory of glasses first at the clustering transition the entropically dominant part of the phase space decomposes into an exponential number of pure states so that beyond this transition a uniform sampling of solutions becomes hard afterward the space of solutions condenses over a finite number of the largest states and consequently the total entropy of solutions becomes smaller than the annealed one another transition takes place when in all the entropically dominant states a finite fraction of nodes freezes so that each of these nodes is allowed a single color in all the solutions inside the state eventually above the coloring threshold no more solutions are available

on the freezing of variables in random constraint satisfaction problems the set of solutions of random constraint satisfaction problems zero energy groundstates of mean field diluted spin glasses undergoes several structural phase transitions as the amount of constraints is increased this set first breaks down into a large number of well separated clusters at the freezing transition which is in general distinct from the clustering one some variables spins take the same value in all solutions of a given cluster in this paper we introduce and study a message passing procedure that allows to compute for generic constraint satisfaction problems the sizes of the rearrangements induced in response to the modification of a variable these sizes diverge at the freezing transition with a critical behavior which is also investigated in details we apply the generic formalism in particular to the random satisfiability of boolean formulas and to the coloring of random graphs the computation is first performed in random tree ensembles for which we underline a connection with percolation models and with the reconstruction problem of information theory the validity of these results for the original random ensembles is then discussed in the framework of the cavity method

optimal stochastic switching under convexity assumptions we address a method of approximate calculation of optimal control policy applicable to a particular class of stochastic control problems whose stochastic dynamics exhibit a certain convexity preserving property problems of this type appear in many applications and encompass important examples arising in the area of optimal stopping and in the framework of control based on partial observations utilizing this specific structure we suggest a numerical method which enjoys a number of desirable properties in particular we work out a remarkably strong method for calculation of the value function within our numerically well trackable approach we show a convergence to the value function of the original problem uniformly on compact sets this issue can be of great advantage particularly for high dimensional control problem where the only competing methods from least squares monte carlo family are able to serve merely l p convergence under several restrictions since the presented algorithm is simpl

random k satisfiability problem from an analytic solution to an efficient algorithm we study the problem of satisfiability of randomly chosen clauses each with k boolean variables using the cavity method at zero temperature we find the phase diagram for the k 3 case we show the existence of an intermediate phase in the satisfiable region where the proliferation of metastable states is at the origin of the slowdown of search algorithms the fundamental order parameter introduced in the cavity method which consists of surveys of local magnetic fields in the various possible states of the system can be computed for one given sample these surveys can be used to invent new types of algorithms for solving hard combinatorial optimizations problems one such algorithm is shown here for the k 3 satisfiability problem with very good performances

structured prediction via output space search we consider a framework for structured prediction based on search in the space of complete structured outputs given a structured input an output is produced by running a time bounded search procedure guided by a learned cost function and then returning the least cost output uncovered during the search this framework can be instantiated for a wide range of search spaces and search procedures and easily incorporates arbitrary structured prediction loss functions in this paper we make two main technical contributions first we describe a novel approach to automatically defining an effective search space over structured outputs which is able to leverage the availability of powerful classification learning algorithms in particular we define the limited discrepancy search space and relate the quality of that space to the quality of learned classifiers we also define a sparse version of the search space to improve the effciency of our overall approach second we give a generic cost function learning approach that is applicable to a wide range of search procedures the key idea is to learn a cost function that attempts to mimic the behavior of conducting searches guided by the true loss function our experiments on six benchmark domains show that a small amount of search in limited discrepancy search space is often sufficient for significantly improving on state of the art structured prediction performance we also demonstrate significant speed improvements for our approach using sparse search spaces with little or no loss in accuracy

supersymmetric field theoretic models on a supermanifold we propose the extension of some structural aspects that have successfully been applied in the development of the theory of quantum fields propagating on a general space time manifold so as to include superfield models on a supermanifold we only deal with the limited class of supermanifolds which admit the existence of a smooth body manifold structure our considerations are based on the catenacci reina teofillatto bryant approach to supermanifolds in particular we show that the class of supermanifolds constructed by bonora pasti tonin satisfies the criteria which guarantee that a supermanifold admits a hausdorff body manifold this construction is the closest to the physicist s intuitive view of superspace as a manifold with some anticommuting coordinates where the odd sector is topologically trivial the paper also contains a new construction of superdistributions and useful results on the wavefront set of such objects moreover a generalization of the spectral condition is formulated using the notion of the wavefront set of superdistributions which is equivalent to the requirement that all of the component fields satisfy on the body manifold a microlocal spectral condition proposed by brunetti fredenhagen kohler

variational calculus and poincare cartan formalism on supermanifolds in this work we show how the poincare cartan form can be used to describe the symmetries of lagrangian supersymmetric possibly p adic field theories we show that the existence of the poincare cartan form in supermanifold theory is ensured only in a relevant class of lagrangian densities moreover we give an abstract characterization of supersymmetric invariance based on the poincare cartan form

learning evaluation functions to improve optimization by local search this paper describes algorithms that learn to improve search performance on large scale optimization tasks the main algorithm stage works by learning an evaluation function that predicts the outcome of a local search algorithm such as hillclimbing or walksat from features of states visited during search the learned evaluation function is then used to bias future search trajectories toward better optima on the same problem another algorithm x stage transfers previously learned evaluation functions to new similar optimization problems empirical results are provided on seven large scale optimization domains bin packing channel routing bayesian network structure finding radiotherapy treatment planning cartogram design boolean satisfiability and boggle board setup

using randomization to break the curse of dimensionality this paper introduces random versions of successive approximations and multigrid algorithms for computing approximate solutions to a class of finite and infinite horizon markovian decision problems mdps we prove that these algorithms succeed in breaking the curse of dimensionality for a subclass of mdps known as discrete decision processes ddps

generalized supermanifolds iii ρ supermanifolds supermanifolds are characterized as supermanifolds modeled in a superspace s having a well behaved body tangent cotangent and tensor bundles to supermanifolds are discussed the elements of the theory of lie supergroups and principal superfiber bundles as supermanifolds are given

instability of one step replica symmetry broken phase in satisfiability problems we reconsider the one step replica symmetry breaking 1rsb solutions of two random combinatorial problems k xorsat and k sat we present a general method for establishing the stability of these solutions with respect to further steps of replica symmetry breaking our approach extends the ideas of montanari and ricci tersenghi 2003 eur phys j b 33 339 to more general combinatorial problems it turns out that 1rsb is always unstable at sufficiently small clause density or high energy in particular the recent 1rsb solution to 3 sat is unstable at zero energy for m with m 4 153 on the other hand the sat unsat phase transition seems to be correctly described within 1rsb