exact thresholds for dpll on random xor sat and np complete extensions of xor sat this paper discusses a model of constraint satisfaction problems known as uniquely extendible constraint satisfaction problems this model includes and generalizes xor sat and the model includes an np complete problem that appears to share many of the threshold characteristics of random sat in this paper we find an exact threshold in the behavior of two versions of dpll on random instances of this problem one version uses the unit clause heuristic and the other uses the generalized unit clause heuristic specifically for dpll with the unit clause heuristic we prove that there is a clause density c smaller than the satisfiability threshold such that for random instances with density smaller than this threshold dpll with unit clause will find a satisfying assignment in linear time with uniformly positive probability however for random instances with density larger than this threshold dpll with unit clause will require exponential time with uniformly positive probability to find a satisfying assignment we then find the equivalent threshold density for dpll with the generalized unit clause heuristic we also prove the analog of the 2 p sat conjecture for this class of problems

on the closest string and substring problems the problem of finding a center string that is close to everygiven string arises in computational molecular biology and codingtheory this problem has two versions the closest string problemand the closest substring problem given a set of strings s s1 s2 sn each of length m the closest stringproblem is to find the smallest d and a string s of lengthm which is within hamming distance d to eachsi e s this problem comes fromcoding theory when we are looking for a code not too far away froma given set of codes closest substring problem with an additionalinput integer l asks for the smallest d and a strings of length l which is within hamming distance daway from a substring of length l of each si this problemis much more elusive than the closest string problem the closestsubstring problem is formulated from applications in findingconserved regions identifying genetic drug targets and generatinggenetic probes in molecular biology whether there are efficientapproximation algorithms for both problems are major open questionsin this area we present two polynomial time approximationalgorithms with approximation ratio 1 e for any smalle to settle both questions

an exactly solvable random satisfiability problem we introduce a new model for the generation of random satisfiability problems it is an extension of the hyper sat model of ricci tersenghi weigt and zecchina which is a variant of the famous k sat model it is extended to q state variables and relates to a different choice of the statistical ensemble the model has an exactly solvable statistic the critical exponents and scaling functions of the sat unsat transition are calculable at zero temperature with no need of replicas also with exact finite size corrections we also introduce an exact duality of the model and show an analogy of thermodynamic properties with the random energy model of disordered spin system theory relations with error correcting codes are also discussed

properties of atypical graphs from negative complexities the one step replica symmetry breaking cavity method is proposed as a new tool to investigate large deviations in random graph ensembles the procedure hinges on a general connection between negative complexities and probabilities of rare samples in spin glass like models this relation between large deviations and replica theory is explicited on different models where it is confronted to direct combinatorial calculations

operator p compact mappings abstract we introduce the classes of operator p compact mappings and completely right p nuclear operators which are natural extensions to the operator space framework of their corresponding banach operator ideals we relate these two classes define natural operator space structures and study several properties of these ideals we show that the class of operator compact mappings in fact coincides with a notion already introduced by webster in the nineties in a very different language this allows us to provide an operator space structure to webster s class

source coding by efficient selection of ground state clusters we analyze the geometrical structure of clusters of ground states which appear in many frustrated systems over random graphs focusing on the regime of connectivities where the number of clusters is exponential in the size of the problems we identify an appropriate generalization of the survey propagation equations efficiently exploring the geometry the possibility of selecting different clusters has also computational consequences as a proof of concept here we show how a well known physical system can be used to perform nontrivial data compression for which we introduce a unique compression scheme performances are optimized when the number of well separated clusters is maximal in the underlying physical model

near optimal configurations in mean field disordered systems we present a general technique to compute how the energy of a configuration varies as a function of its overlap with the ground state in the case of optimization problems our approach is based on a generalization of the cavity method to a system interacting with its ground state with this technique we study the random matching problem as well as the mean field diluted spin glass as a by product of this approach we calculate the de almeida thouless transition line of the spin glass on a fixed connectivity random graph

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

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

statistical mechanics of maximal independent sets the graph theoretic concept of maximal independent set arises in several practical problems in computer science as well as in game theory a maximal independent set is defined by the set of occupied nodes that satisfy some packing and covering constraints it is known that finding minimum and maximum density maximal independent sets are hard optimization problems in this paper we use cavity method of statistical physics and monte carlo simulations to study the corresponding constraint satisfaction problem on random graphs we obtain the entropy of maximal independent sets within the replica symmetric and one step replica symmetry breaking frameworks shedding light on the metric structure of the landscape of solutions and suggesting a class of possible algorithms this is of particular relevance for the application to the study of strategic interactions in social and economic networks where maximal independent sets correspond to pure nash equilibria of a graphical game of public goods allocation

new criterion of the approximation property this paper is concerned with the approximation property which is an important property in banach space theory we show that a banach space x has the approximation property if and only if for every banach space y the set of finite rank operators from x to y is dense in the corresponding space of compact operators in the usual topology of uniform convergence on compact sets

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

a remark on the approximation of p compact operators by finite rank operators abstract the banach operator ideal k p of p compact operators was introduced in p d sinha a k karn compact operators whose adjoints factor through subspaces of l p studia math 150 2002 17 33 let p 1 and let x be a closed subspace of an l p space we show that x has the approximation property if and only if for every banach space y the linear space f y x of finite rank operators is k p norm dense in k p y x i e x has the k p approximation property

statistical mechanics of the hitting set problem in this paper we present a detailed study of the hitting set hs problem this problem is a generalization of the standard vertex cover to hypergraphs one seeks a configuration of particles with minimal density such that every hyperedge of the hypergraph contains at least one particle it can also be used in important practical tasks such as the group testing procedures where one wants to detect defective items in a large group by pool testing using a statistical mechanics approach based on the cavity method we study the phase diagram of the hs problem in the case of random regular hypergraphs depending on the values of the variables and tests degrees different situations can occur the hs problem can be either in a replica symmetric phase or in a one step replica symmetry breaking one in these two cases we give explicit results on the minimal density of particles and the structure of the phase space these problems are thus in some sense simpler than the original vertex cover problem where the need for a full replica symmetry breaking has prevented the derivation of exact results so far finally we show that decimation procedures based on the belief propagation and the survey propagation algorithms provide very efficient strategies to solve large individual instances of the hitting set problem

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

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

the dual space of l x y τp and the p approximation property we establish a representation of the dual space of l x y the space of bounded linear operators from a banach space x into a banach space y endowed with the topology p of uniform convergence on p compact subsets of x we apply this representation and solve the duality problem for the p approximation property p ap that is if the dual space x has the p ap then so does x however the converse does not hold in general we show that given 2 2p p 2 this subspace is the davie space in lq davie 1973 5 which does not have the approximation property it follows that for every 2 p there exists a banach space yp such that it has the p ap but its dual space yp fails to have the p ap we study the relation of the p ap with the denseness of finite rank operators in the topology p finally we introduce the p compact approximation property p cap and show for every 2 p that the davie space in c0 fails to have the p cap and also that a variant of the willis space willis 1992 17 has the p cap but it fails to have the p ap

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

efficient algorithms for consensus string problems minimizing both distance sum and radius abstract the consensus string problem is finding a representative string called a consensus of a given set s of strings in this paper we deal with consensus problems considering both distance sum and radius where the distance sum is the sum of hamming distances from the strings in s to the consensus and the radius is the longest hamming distance from the strings in s to the consensus although there have been results considering either distance sum or radius there have been no results considering both to the best of our knowledge we present the first algorithms for two consensus problems considering both distance sum and radius for three strings one problem is to find an optimal consensus minimizing both distance sum and radius the other problem is to find a bounded consensus such that the distance sum is at most s and the radius is at most r for given constants s and r our algorithms are based on characterization of the lower bounds of distance sum and radius and thus they solve the problems efficiently both algorithms run in linear time

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

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

inference by replication in densely connected systems an efficient bayesian inference method for problems that can be mapped onto dense graphs is presented the approach is based on message passing where messages are averaged over a large number of replicated variable systems exposed to the same evidential nodes an assumption about the symmetry of the solutions is required for carrying out the averages here we extend the previous derivation based on a replica symmetric rs like structure to include a more complex one step replica symmetry breaking like 1rsb like ansatz to demonstrate the potential of the approach it is employed for studying critical properties of the ising linear perceptron and for multiuser detection in code division multiple access cdma under different noise models results obtained under the rs assumption in the noncritical regime give rise to a highly efficient signal detection algorithm in the context of cdma while in the critical regime one observes a first order transition line that ends in a continuous phase transition point finite size effects are also observed while the 1rsb ansatz is not required for the original problems it was applied to the cdma signal detection problem with a more complex noise model that exhibits rsb behavior resulting in an improvement in performance

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

a modified belief propagation algorithm for code word quantization modern coding advances including dirty paper coding and information hiding require quantizing a given binary word to a code word a good solution would approach the rate distortion bound in lossy source compression here we propose a simple variant on belief propagation which is observed to converge to a solution giving respectable rate distortion performance comparisons with other recently proposed source quantization methods reveal that the proposed algorithm holds particular interest in short block length applications as encountered in packet based communication systems

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

what makes a phase transition analysis of the random satisfiability problem abstract in the last 30 years it was found that many combinatorial systems undergo phase transitions one of the most important examples of these can be found among the random k satisfiability problems often referred to as k sat asking whether there exists an assignment of boolean values satisfying a boolean formula composed of clauses with k random variables each the random 3 sat problem is reported to show various phase transitions at different critical values of the ratio of the number of clauses to the number of variables the most famous of these occurs when the probability of finding a satisfiable instance suddenly drops from 1 to 0 this transition is associated with a rise in the hardness of the problem but until now the correlation between any of the proposed phase transitions and the hardness is not totally clear in this paper we will first show numerically that the number of solutions universally follows a lognormal distribution thereby explaining the puzzling question of why the number of solutions is still exponential at the critical point moreover we provide evidence that the hardness of the closely related problem of counting the total number of solutions does not show any phase transition like behavior this raises the question of whether the probability of finding a satisfiable instance is really an order parameter of a phase transition or whether it is more likely to just show a simple sharp threshold phenomenon more generally this paper aims at starting a discussion where a simple sharp threshold phenomenon turns into a genuine phase transition

statistical physics of loopy interactions independent loop approximation and beyond we consider an interacting system of spin variables on a loopy interaction graph identified by a tree graph and a set of loopy interactions we start from a high temperature expansion for loopy interactions represented by a sum of non negative contributions from all the possible frustration free loop configurations we then compute the loop corrections using different approximations for the nonlocal loop interactions induced by the spin correlations in the tree graph for distant loopy interactions we can exploit the exponential decay of correlations in the tree interaction graph to compute loop corrections within an independent loop approximation higher orders of the approximation are obtained by considering the correlations between the nearby loopy interactions involving larger number of spin variables in particular the sum over the loop configurations can be computed exactly by the belief propagation algorithm in the low orders of the approximation as long as the loopy interactions have a tree structure these results might be useful in developing more accurate and convergent message passing algorithms exploiting the structure of loopy interactions

tap complexity the cavity method and supersymmetry we compute the bray and moore bm tap complexity for the sherrington kirkpatrick model through the cavity method showing that some essential modifications are needed with respect to the standard formulation of the method this allows us to understand various features recently discovered and to unveil at last the physical meaning of the parameters of the bm theory we also reconsider the supersymmetric susy formulation of the problem finding that the bm solution satisfies some proper susy ward identities that are different from the standard ones the susy relationships encode the physical meaning of the parameters obtained through the cavity method the problem of the vanishing prefactor is addressed showing how it can be avoided

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

statistical mechanics of optimization problems abstract here i will present an introduction to the results that have been recently obtained in constraint optimization of random problems using statistical mechanics techniques after presenting the general results in order to simplify the presentation i will describe in details the problems related to the coloring of a random graph

bicolouring random hypergraphs we study the problem of bicolouring random hypergraphs both numerically and analytically we apply the zero temperature cavity method to find analytical results for the phase transitions dynamic and static in the one step replica symmetry breaking 1rsb approximation these points appear to be in agreement with the results of the numerical algorithm in the second part we implement and test the survey propagation algorithm for specific bicolouring instances in the so called hard sat phase

cavity approach to sphere packing in hamming space in this paper we study the hard sphere packing problem in the hamming space by the cavity method we show that both the replica symmetric and the replica symmetry breaking approximations give maximum rates of packing that are asymptotically the same as the lower bound of gilbert and varshamov consistently with known numerical results the replica symmetric equations also suggest a crystalline solution where for even diameters the spheres are more likely to be found in one of the subspaces even or odd of the hamming space these crystalline packings can be generated by a recursive algorithm which finds maximum packings in an ultra metric space finally we design a message passing algorithm based on the cavity equations to find dense packings of hard spheres known maximum packings are reproduced efficiently i n non trivial ranges of dimensions and number of spheres

on α nuclear operators with applications to vector valued function spaces abstract let x be a banach space and let y be a closed subspace of a banach space z let be a tensor norm our main result is as follows assume that x or z has the approximation property if there is a bounded linear extension operator from y to z then any bounded linear operator t x y is nuclear whenever t is nuclear from x to z using this result we characterize the space c p x respectively uc p x of continuous functions from a compact hausdorff space into a banach space x whose range is p compact respectively unconditionally p compact

alternative variational approach to cactus lattices in this paper i will present an alternative approach to the bethe or cactus lattice approximation widely employed in the theory of cooperative phenomena this approach relies on a variational free energy which is equivalent to the bethe free energy in that it has the same stationary points but allows one to simplify analytical calculations since it is a function of only single site probability distributions in the same way as an ordinary mean field bragg williams free energy as an application i shall discuss a derivation of closed form equations for critical points in ising like models moreover i will suggest a rule of thumb to choose the cactus lattice connectivity yielding the best approximation for the corresponding model defined on an ordinary lattice

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

spin glass phase transitions and minimum energy of the random feedback vertex set problem a feedback vertex set fvs of an undirected graph contains vertices from every cycle of this graph constructing a fvs of sufficiently small cardinality is very difficult in the worst cases but for random graphs this problem can be efficiently solved by converting it into an appropriate spin glass model h j zhou eur phys j b 86 455 2013 in the present work we study the spin glass phase transitions and the minimum energy density of the random fvs problem by the first step replica symmetry breaking 1rsb mean field theory for both regular random graphs and erdos renyi graphs we determine the inverse temperature beta 1 at which the replica symmetric mean field theory loses its local stability the inverse temperature beta d of the dynamical clustering phase transition and the inverse temperature beta s of the static condensation phase transition these critical inverse temperatures all change with the mean vertex degree in a nonmonotonic way and beta d is distinct from beta s for regular random graphs of vertex degrees k 60 while beta d are identical to beta s for erdos renyi graphs at least up to mean vertex degree c 512 we then derive the zero temperature limit of the 1rsb theory and use it to compute the minimum fvs cardinality

minimal contagious sets in random regular graphs the bootstrap percolation or threshold model is a dynamic process modelling the propagation of an epidemic on a graph where inactive vertices become active if their number of active neighbours reach some threshold we study an optimization problem related to it namely the determination of the minimal number of active sites in an initial configuration that leads to the activation of the whole graph under this dynamics with and without a constraint on the time needed for the complete activation this problem encompasses in special cases many extremal characteristics of graphs like their independence decycling or domination number and can also be seen as a packing problem of repulsive particles we use the cavity method including the effects of replica symmetry breaking an heuristic technique of statistical mechanics many predictions of which have been confirmed rigorously in the recent years we have obtained in this way several quantitative conjectures on the size of minimal contagious sets in large random regular graphs the most striking being that 5 regular random graph with a threshold of activation of 3 resp 6 regular with threshold 4 have contagious sets containing a fraction 1 6 resp 1 4 of the total number of vertices equivalently these numbers are the minimal fraction of vertices that have to be removed from a 5 regular resp 6 regular random graph to destroy its 3 core we also investigated survey propagation like algorithmic procedures for solving this optimization problem on single instances of random regular graphs

generalized string matching given a pattern string of length n and an object string of length m the string matching problem asks for the positions of all occurrences of the pattern in the object string this paper investigates a generalization of string matching in which the pattern is a sequence of pattern elements each compatible with a set of symbols the alphabet of symbols is infinite with its members encoded in a finite alphabet in contrast to standard string matching which can be solved in simultaneous linear time and constant space it is shown that generalized string matching requires a time space product of omega n 2 log n on a powerful model of computation when the alphabet is restricted to n symbols our proof uses a method of borodin the obvious algorithm for generalized string matching requires time o nm where n is the length of the encoding of the pattern and m is that of the object string we describe an algorithm which solves generalized string matching in time o n m mn 1 2 o

on p compact mappings and the p approximation property abstract the notion of p compact sets arises naturally from grothendieck s characterization of compact sets as those contained in the convex hull of a norm null sequence the definition due to sinha and karn 2002 leads to the concepts of p approximation property and p compact operators which form an ideal with its ideal norm p this paper examines the interaction between the p approximation property and certain space of holomorphic functions the p compact analytic functions in order to understand these functions we define a p compact radius of convergence which allows us to give a characterization of the functions in the class we show that p compact holomorphic functions behave more like nuclear than compact maps we use the product of schwartz to characterize the p approximation property of a banach space in terms of p compact homogeneous polynomials and in terms of p compact holomorphic functions with range on the space finally we show that p compact holomorphic functions fit into the framework of holomorphy types which allows us to inspect the p approximation property our approach also allows us to solve several questions posed by aron maestre and rueda 2010

hiding quiet solutions in random constraint satisfaction problems we study constraint satisfaction problems on the so called planted random ensemble we show that for a certain class of problems e g graph coloring many of the properties of the usual random ensemble are quantitatively identical in the planted random ensemble we study the structural phase transitions and the easy hard easy pattern in the average computational complexity we also discuss the finite temperature phase diagram finding a close connection with the liquid glass solid phenomenology

scaling analysis of affinity propagation we analyze and exploit some scaling properties of the em affinity propagation ap clustering algorithm proposed by frey and dueck 2007 following a divide and conquer strategy we setup an exact renormalization based approach to address the question of clustering consistency in particular how many cluster are present in a given data set we first observe that the divide and conquer strategy used on a large data set hierarchically reduces the complexity cal o n 2 to cal o n h 2 h 1 for a data set of size n and a depth h of the hierarchical strategy for a data set embedded in a d dimensional space we show that this is obtained without notably damaging the precision except in dimension d 2 in fact for d larger than 2 the relative loss in precision scales like n 2 d h 1 d finally under some conditions we observe that there is a value s of the penalty coefficient a free parameter used to fix the number of clusters which separates a fragmentation phase for ss of the underlying hidden cluster structure at this precise point holds a self similarity property which can be exploited by the hierarchical strategy to actually locate its position as a result of an exact decimation procedure from this observation a strategy based on ap can be defined to find out how many clusters are present in a given dataset

inference algorithm for finite dimensional spin glasses belief propagation on the dual lattice starting from a cluster variational method and inspired by the correctness of the paramagnetic ansatz at high temperatures in general and at any temperature in the two dimensional 2d edwards anderson ea model we propose a message passing algorithm the dual algorithm to estimate the marginal probabilities of spin glasses on finite dimensional lattices we use the ea models in 2d and 3d as benchmarks the dual algorithm improves the bethe approximation and we show that in a wide range of temperatures compared to the bethe critical temperature our algorithm compares very well with monte carlo simulations with the double loop algorithm and with exact calculation of the ground state of 2d systems with bimodal and gaussian interactions moreover it is usually 100 times faster than other provably convergent methods as the double loop algorithm in 2d and 3d the quality of the inference deteriorates only where the correlation length becomes very large i e at low temperatures in 2d and close to the critical temperature in 3d

coloring random graphs we study the graph coloring problem over random graphs of finite average connectivity c given a number q of available colors we find that graphs with low connectivity admit almost always a proper coloring whereas graphs with high connectivity are uncolorable depending on q we find the precise value of the critical average connectivity c q moreover we show that below c q there exists a clustering phase c e c d c q in which ground states spontaneously divide into an exponential number of clusters and where the proliferation of metastable states is responsible for the onset of complexity in local search algorithms

replica bounds for diluted non poissonian spin systems in this paper we extend replica bounds and free energy subadditivity arguments to diluted spin glass models on graphs with arbitrary non poissonian degree distribution the new difficulties specific of this case are overcome introducing an interpolation procedure that stresses the relation between interpolation methods and the cavity method as a byproduct we obtain self averaging identities that generalize the ghirlanda guerra ones to the multi overlap case

survey propagation at finite temperature application to a sourlas code as a toy model in this paper we investigate a finite temperature generalization of survey propagation by applying it to the problem of finite temperature decoding of a biased finite connectivity sourlas code for temperatures lower than the nishimori temperature we observe that the result is a shift of the location of the dynamical critical channel noise to larger values than the corresponding dynamical transition for belief propagation as suggested recently by migliorini and saad for ldpc codes we show how the finite temperature 1rsb sp gives accurate results in the regime where competing approaches fail to converge or fail to recover the retrieval state

minimizing energy below the glass thresholds focusing on the optimization version of the random k satisfiability problem the max k sat problem we study the performance of the finite energy version of the survey propagation algorithm we show that a simple linear time backtrack decimation strategy is sufficient to reach configurations well below the lower bound for the dynamic threshold energy and very close to the analytic prediction for the optimal ground states a comparative numerical study on one of the most efficient local search procedures is also given

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

phase transition for cutting plane approach to vertex cover problem we study the vertex cover problem which is a nondeterministic polynomial time hard optimization problem and a prototypical model exhibiting phase transitions on random graphs such as erd s renyi er random graphs these phase transitions coincide with changes of the solution space structure e g for the er ensemble at connectivity c e 2 7183 from replica symmetric to replica symmetry broken for the vertex cover problem the typical complexity of exact branch and bound algorithms which proceed by exploring the landscape of feasible configurations also changes close to this phase transition from easy to hard in this work we consider an algorithm which has a completely different strategy the problem is mapped onto a linear programming problem augmented by a cutting plane approach hence the algorithm operates in a space outside the space of feasible configurations until the final step where a solution is found here we show that this type of algorithm also exhibits an easy hard transition around c e which strongly indicates that the typical hardness of a problem is fundamental to the problem and not due to a specific representation of the problem

on the survey propagation equations in random constraint satisfiability problems in this paper we study the existence of a solution of the survey propagation equations for a given instance of a random problem in the framework of constraint satisfiability we consider the concrete examples of k satisfiability and coloring we conjecture that when the number of variables goes to infinity the solution of the survey propagation equations for a given instance can be obtained by finding the supposed unique solution of the corresponding equations on an infinite tree we conjecture that the survey propagation equations on a random infinite tree have a unique solution in the suitable range of parameters we also present analytic arguments that indicate that the survey propagation equations do have solutions in the satisfiable phase for simplicity of notation the argument is presented in the case of the coloring problem the same results extend to other optimization problems where exist configurations that have cost zero i e in the satisfiable phase on a random graph the solutions of th

cluster variation method in statistical physics and probabilistic graphical models the cluster variation method cvm is a hierarchy of approximate variational techniques for discrete ising like models in equilibrium statistical mechanics improving on the mean field approximation and the bethe peierls approximation which can be regarded as the lowest level of the cvm in recent years it has been applied both in statistical physics and to inference and optimization problems formulated in terms of probabilistic graphical models the foundations of the cvm are briefly reviewed and the relations with similar techniques are discussed the main properties of the method are considered with emphasis on its exactness for particular models and on its asymptotic properties the problem of the minimization of the variational free energy which arises in the cvm is also addressed and recent results about both provably convergent and message passing algorithms are discussed

finding consensus and optimal alignment of circular strings circular strings are different from linear strings in that the last symbol is considered to precede the first symbol even though circular strings are biologically important only a few efforts have been made to solve computational problems on circular strings in this paper we introduce consensus problems for circular strings of length n and present the first non trivial algorithms to find a consensus and an optimal alignment for circular strings by the hamming distance they are o n 2logn time algorithms for three circular strings and an o n 3logn time algorithm for four circular strings our algorithms are o n logn times faster than the naive algorithms directly using the solutions for the linear consensus problems which take o n 3 time for three circular strings and o n 4 time for four circular strings this speedup was achieved by reducing the problems into correlations and by formulating and solving systems of linear equations moreover our algorithms use only o n space

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

replica cluster variational method we present a general formalism to make the replica symmetric and replica symmetry breaking ansatz in the context of kikuchi s cluster variational method cvm using replicas and the message passing formulation of cvm we obtain a variational expression of the replicated free energy of a system with quenched disorder both averaged and on a single sample and make the hierarchical ansatz using functionals of functions of fields to represent the messages we obtain a set of integral equations for the message functionals the main difference with the bethe case is that the functionals appear in the equations in implicit form and are not positive definite thus standard iterative population dynamic algorithms cannot be used to determine them in the simplest cases the solution could be obtained iteratively using fourier transforms

lossy source compression using low density generator matrix codes analysis and algorithms we study the use of low density generator matrix ldgm codes for lossy compression of the bernoulli symmetric source first we establish rigorous upper bounds on the average distortion achieved by check regular ensemble of ldgm codes under optimal minimum distance source encoding these bounds establish that the average distortion using such bounded degree families rapidly approaches the shannon limit as the degrees are increased second we propose a family of message passing algorithms ranging from the standard belief propagation algorithm at one extreme to a variant of survey propagation algorithm at the other when combined with a decimation subroutine and applied to ldgm codes with suitably irregular degree distributions we show that such a message passing decimation algorithm yields distortion very close to the shannon rate distortion bound for the binary symmetric source

uniform factorization for p compact sets of p compact linear operators abstract obtaining a factorization of p compact linear operators via universal banach spaces and using the lifting property of quotient maps for p compact sets we prove a factorization result for relatively r compact subsets of p compact operators where r 2 1 p r to apply our results to homogeneous polynomials in particular we show that relatively p compact subsets of a banach space of p compact operators are collectively p compact

some spin glass ideas applied to the clique problem in this paper we introduce a new algorithm to study some np complete problems this algorithm is a markov chain monte carlo mcmc inspired by the cavity method developed in the study of spin glass we will focus on the maximum clique problem and we will compare this new algorithm with several standard algorithms on some dimacs benchmark graphs and on random graphs the performances of the new algorithm are quite surprising our effort in this paper is to be clear as well to those readers who are not in the field

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

the p approximation property in terms of density of finite rank operators abstract we characterize the p approximation property p ap introduced by sinha and karn d p sinha a k karn compact operators whose adjoints factor through subspaces of l p studia math 150 2002 17 33 in terms of density of finite rank operators in the spaces of p compact and of adjoints of p summable operators as application the p ap of dual banach spaces is characterized via density of finite rank operators in the space of quasi p nuclear operators this relates the p ap to saphar s approximation property ap p as another application the p ap is characterized via a trace condition allowing to define the trace functional on certain subspaces of the space of nuclear operators

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

finding long cycles in graphs we analyze the problem of discovering long cycles inside a graph we propose and test two algorithms for this task the first one is based on recent advances in statistical mechanics and relies on a message passing procedure the second follows a more standard monte carlo markov chain strategy special attention is devoted to hamiltonian cycles of nonregular random graphs of minimal connectivity equal to 3

on the satisfiability threshold of formulas with three literals per clause in this paper we present a new upper bound for randomly chosen 3 cnf formulas in particular we show that any random formula over n variables with a clauses to variables ratio of at least 4 4898 is as n grows large asymptotically almost surely unsatisfiable the previous best such bound due to dubois in 1999 was 4 506 the first such bound independently discovered by many groups of researchers since 1983 was 5 19 several decreasing values between 5 19 and 4 506 were published in the years between we believe that the probabilistic techniques we use for the proof are of independent interest

witness of unsatisfiability for a random 3 satisfiability formula the random 3 satisfiability 3 sat problem is in the unsatisfiable unsat phase when the clause density alpha exceeds a critical value alpha s approximate to 4 267 rigorously proving the unsatisfiability of a given large 3 sat instance is however extremely difficult in this paper we apply the mean field theory of statistical physics to the unsatisfiability problem and show that a reduction to 3 xorsat which permits the construction of a specific type of unsat witnesses feige kim ofek witnesses is possible when the clause density alpha 19 we then construct feige kim ofek witnesses for single 3 sat instances through a simple random sampling algorithm and a focused local search algorithm the random sampling algorithm works only when a scales at least linearly with the variable number n but the focused local search algorithm works for clause density alpha cn b with b approximate to 0 59 and prefactor c approximate to 8 the exponent b can be further decreased by enlarging the single parameter s of the focused local search algorithm

a note on p limited sets abstract given p 1 a subset a of a banach space x is said to be p limited if for every weakly p summable sequence x n in x there exists n l p such that x n x n for all x a and n n it is showed that p limited sets are q limited whenever p q and banach spaces enjoying the property that every q limited subset is p limited are characterized we also prove that an operator has p summing adjoint if and only if it maps relatively compact sets to p limited sets

stability analysis on the finite temperature replica symmetric and first step replica symmetry broken cavity solutions of the random vertex cover problem the vertex cover problem is a prototypical hard combinatorial optimization problem it was studied in recent years by physicists using the cavity method of statistical mechanics in this paper the stability of the finite temperature replica symmetric rs and the first step replica symmetry broken 1rsb cavity solutions of the vertex cover problem on random regular graphs of finite vertex degree k are analyzed by population dynamics simulations we found that 1 the lowest temperature for the rs solution to be stable t rs k is not a monotonic function of k 2 at relatively large connectivity k and temperature t slightly below the dynamic transition temperature t d k the 1rsb solutions with small but non negative complexity values are stable and 3 the dynamical transition temperature t d and kauzmann temperature t k is equal to each other similar results are obtained on random poissonian graphs

the banach ideal of a compact operators and related approximation properties abstract we use the notion of a compact sets determined by an operator ideal a introduced by carl and stephani 1984 to show that many known results of certain approximation properties and several ideals of compact operators can be systematically studied under this framework for banach operator ideals a we introduce a way to measure the size of a compact sets and use it to give a norm on k a the ideal of a compact operators then we study two types of approximation properties determined by a compact sets we focus our attention on an approximation property which makes use of the norm defined on k a this notion fits the definition of the a approximation property recently introduced by oja 2012 with k a instead of a we exemplify the power of the carl stephani theory and the geometric structure introduced here by appealing to some recent developments on p compactness