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 constraint satisfaction easy generation of hard satisfiable instances in this paper we show that the models of random csp instances proposed by xu and li k xu w li exact phase transitions in random constraint satisfaction problems journal of artificial intelligence research 12 2000 93 103 k xu w li many hard examples in exact phase transitions with application to generating hard satisfiable instances technical report corr report cs cc 0302001 revised version in theoretical computer science 355 2006 291 302 are of theoretical and practical interest indeed these models called rb and rd present several nice features first it is quite easy to generate random instances of any arity since no particular structure has to be integrated or property enforced in such instances then the existence of an asymptotic phase transition can be guaranteed while applying a limited restriction on domain size and on constraint tightness in that case a threshold point can be precisely located and all instances have the guarantee to be hard at the threshold i e to have an exponential tree resolution complexity next a formal analysis shows that it is possible to generate forced satisfiable instances whose hardness is similar to unforced satisfiable ones this analysis is supported by some representative results taken from an intensive experimentation that we have carried out using complete and incomplete search methods

an optimal coarse grained arc consistency algorithm the use of constraint propagation is the main feature of any constraint solver it is thus of prime importance to manage the propagation in an efficient and effective fashion there are two classes of propagation algorithms for general constraints fine grained algorithms where the removal of a value for a variable will be propagated to the corresponding values for other variables and coarse grained algorithms where the removal of a value will be propagated to the related variables one big advantage of coarse grained algorithms like ac 3 over fine grained algorithms like ac 4 is the ease of integration when implementing an algorithm in a constraint solver however fine grained algorithms usually have optimal worst case time complexity while coarse grained algorithms do not for example ac 3 is an algorithm with non optimal worst case complexity although it is simple efficient in practice and widely used in this paper we propose a coarse grained algorithm ac2001 3 1 that is worst case optimal and preserves as much as possible the ease of its integration into a solver no heavy data structure to be maintained during search experimental results show that ac2001 3 1 is competitive with the best fine grained algorithms such as ac 6 the idea behind the new algorithm can immediately be applied to obtain a path consistency algorithm that has the best known time and space complexity the same idea is then extended to non binary constraints

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

ground state entropy of the random vertex cover problem counting the number of ground states for a spin glass or nondeterministic polynomial complete combinatorial optimization problem is even more difficult than the already hard task of finding a single ground state in this paper the entropy of minimum vertex covers of random graphs is estimated through a set of iterative equations based on the cavity method of statistical mechanics during the iteration both the cavity entropy contributions and cavity magnetizations for each vertex are updated this approach overcomes the difficulty of iterative divergence encountered in the zero temperature first step replica symmetry breaking 1rsb spin glass theory it is still applicable when the 1rsb mean field theory is no longer stable the method can be extended to compute the entropies of ground states and metastable minimal energy states for other random graph spin glass systems

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

a general model and thresholds for random constraint satisfaction problems in this paper we study the relation among the parameters in their most general setting that define a large class of random csp models d k csp where d is the domain size and k is the length of the constraint scopes the model d k csp unifies several related models such as the model rb and the model k csp we prove that the model d k csp exhibits exact phase transitions if klnd increases no slower than the logarithm of the number of variables a series of experimental studies with interesting observations are carried out to illustrate the solubility phase transition and the hardness of instances around phase transitions

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

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

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

looking for symmetric bell inequalities finding all bell inequalities for a given number of parties measurement settings and measurement outcomes is in general a computationally hard task we show that all bell inequalities which are symmetric under the exchange of parties can be found by examining a symmetrized polytope which is simpler than the full bell polytope as an illustration of our method we generate 238 885 new bell inequalities and 1085 new svetlichny inequalities we find in particular facet inequalities for bell experiments involving two parties and two measurement settings that are not of the collins gisin linden massar popescu type

reasoning from last conflict s in constraint programming constraint programming is a popular paradigm to deal with combinatorial problems in artificial intelligence backtracking algorithms applied to constraint networks are commonly used but suffer from thrashing i e the fact of repeatedly exploring similar subtrees during search an extensive literature has been devoted to prevent thrashing often classified into look ahead constraint propagation and search heuristics and look back intelligent backtracking and learning approaches in this paper we present an original look ahead approach that allows to guide backtrack search toward sources of conflicts and as a side effect to obtain a behavior similar to a backjumping technique the principle is the following after each conflict the last assigned variable is selected in priority so long as the constraint network cannot be made consistent this allows us to find following the current partial instantiation from the leaf to the root of the search tree the culprit decision that prevents the last variable from being assigned this way of reasoning can easily be grafted to many variations of backtracking algorithms and represents an original mechanism to reduce thrashing moreover we show that this approach can be generalized so as to collect a small set of incompatible variables that are together responsible for the last conflict experiments over a wide range of benchmarks demonstrate the effectiveness of this approach in both constraint satisfaction and automated artificial intelligence planning

statistical mechanics of the minimum vertex cover problem in stochastic block models the minimum vertex cover min vc problem is a well known np hard problem earlier studies illustrate that the problem defined over the erdos renyi random graph with a mean degree c exhibits computational difficulty in searching the min vc set above a critical point c e 2 718 here we address how this difficulty is influenced by the mesoscopic structures of graphs for this we evaluate the critical condition of difficulty for the stochastic block model we perform a detailed examination of the specific cases of two equal size communities characterized by in and out degrees which are denoted by c in and c out respectively our analysis based on the cavity method indicates that the solution search once becomes difficult when c in c out exceeds e from below but becomes easy again when c out is sufficiently larger than c in in the region c out e experiments based on various search algorithms support the theoretical prediction

spin glass model for the c dismantling problem c dismantling cd problem aims at finding the minimum vertex set d of a graph g v e after removing which the remaining graph will break into connected components with the size not larger than c in this paper we introduce a spin glass model with c 1 integer value states into the cd problem and then study the properties of this spin glass model by the belief propagation bp equations under the replica symmetry ansatz we give the lower bound rho c of the relative size of d with finite c on regular random graphs and erdos renyi random graphs we find rho c will decrease gradually with growing c and it converges to rho infty as c to infty the cd problem is called dismantling problem when c is a small finite fraction of v therefore rho infty is also the lower bound of the dismantling problem when v to infty to reduce the computation complexity of the bp equations taking the knowledge of the probability of a random selected vertex belonging to a remaining connected component with the size a the original bp equations can be simplified to one with only three states when c to infty the simplified bp equations are very similar to the bp equations of the feedback vertex set spin glass model h j zhou eur phys j b 86 455 2013 at last we develop two practical belief propagation guide decimation algorithms based on the original bp equations cd bpd and the simplified bp equations scd bpd to solve the cd problem on a certain graph our bpd algorithms and two other state of art heuristic algorithms are applied on various random graphs and some real world networks computation results show that the cd bpd is the best in all tested algorithms in the case of small c but considering the performance and computation consumption we recommend using scd bpd for the network with small clustering coefficient when c is large

exhaustive enumeration unveils clustering and freezing in the random 3 satisfiability problem we study geometrical properties of the complete set of solutions of the random 3 satisfiability problem we show that even for moderate system sizes the number of clusters corresponds surprisingly well with the theoretic asymptotic prediction we locate the freezing transition in the space of solutions which has been conjectured to be relevant in explaining the onset of computational hardness in random constraint satisfaction problems

a framework for the study of symmetric full correlation bell like inequalities full correlation bell like inequalities represent an important subclass of bell like inequalities that have found applications in both a better understanding of fundamental physics and in quantum information science loosely speaking these are inequalities where only measurement statistics involving all parties play a role in this paper we provide a framework for the study of a large family of such inequalities that are symmetrical with respect to arbitrary permutations of the parties as an illustration of the power of our framework we derive i a new family of svetlichny inequalities for arbitrary numbers of parties settings and outcomes ii a new family of two outcome device independent entanglement witnesses for genuine n partite entanglement and iii a new family of two outcome tsirelson inequalities for arbitrary numbers of parties and settings we also discuss briefly the application of these new inequalities in the characterization of quantum correlations

multipartite nonlocality as a resource and quantum correlations having indefinite causal order the characterization of quantum correlations in terms of information theoretic resource has been a fruitful approach to understand the power of quantum correlations as a resource while bipartite entanglement and bell inequality violation in this setting have been extensively studied relatively little is known about their multipartite counterpart in this paper we apply and adapt the recently proposed definitions of multipartite nonlocality bancal et al 2013 phys rev a 88 014102 to the three and four partite scenarios to gain new insight on the resource aspect of multipartite nonlocal quantum correlations specifically we show that reproducing certain tripartite quantum correlations requires mixtures of classical resources be it the ability to change the groupings or the time orderings of measurements thus when seen from the perspective of biseparable one way classical signaling resources certain tripartite quantum correlations do not admit a definite causal order in the four partite scenario we obtain a superset description of the set of biseparable correlations which can be produced by allowing two groups of bipartite non signaling resources quantum violation of the resulting bell like inequalities is investigated as a byproduct we obtain some new examples of device independent witnesses for genuine four partite entanglement and also device independent witnesses that allow one to infer the structure of the underlying multipartite entanglement this article is part of a special issue of journal of physics a mathematical and theoretical devoted to 50 years of bell s theorem

bounded universal functions in one and several complex variables we show how to obtain functions that are universal for the ball of h infty omega where omega subset mathbb c n the existence of our functions will follow from universality criteria but we also show how to construct them then we study the connection between certain interpolating sequences runaway automorphisms and the existence of universal functions on domains in mathbb c n

minimal vertex covers on finite connectivity random graphs a hard sphere lattice gas picture the minimal vertex cover or maximal independent set problem is studied on random graphs of finite connectivity analytical results are obtained by a mapping to a lattice gas of hard spheres of chemical radius 1 and they are found to be in excellent agreement with numerical simulations we give a detailed description of the replica symmetric phase including the size and entropy of the minimal vertex covers and the structure of the unfrozen component which is found to percolate at a connectivity c approximately 1 43 the replica symmetric solution breaks down at c e approximately 2 72 we give a simple one step replica symmetry broken solution and discuss the problems in the interpretation and generalization of this solution

bounding the set of quantum correlations we introduce a hierarchy of conditions necessarily satisfied by any distribution p ensuremath alpha ensuremath beta representing the probabilities for two separate observers to obtain outcomes ensuremath alpha and ensuremath beta when making local measurements on a shared quantum state each condition in this hierarchy is formulated as a semidefinite program among other applications our approach can be used to obtain upper bounds on the quantum violation of an arbitrary bell inequality it yields for instance tight bounds for the violations of the collins et al inequalities

short proofs are narrow resolution made simple the width of a resolution proof is defined to be the maximal number of literals in any clause of the proof in this paper we relate proof width to proof length size in both general resolution and its tree like variant the following consequences of these relations reveal width as a crucial resource of resolution proofs in one direction the relations allow us to give simple unified proofs for almost all known exponential lower bounds on size of resolution proofs as well as several interesting new ones they all follow from width lower bounds and we show how these follow from natural expansion property of clauses of the input tautology in the other direction the width size relations naturally suggest a simple dynamic programming procedure for automated theorem proving one which simply searches for small width proofs this relation guarantees that the runnuing time and thus the size of the produced proof is at most quasi polynomial in the smallest tree like proof this algorithm is never much worse than any of the recursive automated provers such as dll used in practice in contrast we present a family of tautologies on which it is exponentially faster

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

bell type inequalities for partial separability in n particle systems and quantum mechanical violations we derive n particle bell type inequalities under the assumption of partial separability i e that the n particle system is composed of subsystems which may be correlated in any way e g entangled but which are uncorrelated with respect to each other these inequalities provide upon violation experi mentally accessible sufficient conditions for full n particle entanglement i e for n particle entanglement that cannot be reduced to mixtures of states in which a smaller number of particles are entangled the inequalities are shown to be maximally violated by the n particle greenberger horne zeilinger states

locating the phase transition in binary constraint satisfaction problems abstract the phase transition in binary constraint satisfaction problems i e the transition from a region in which almost all problems have many solutions to a region in which almost all problems have no solutions as the constraints become tighter is investigated by examining the behaviour of samples of randomly generated problems in contrast to theoretical work which is concerned with the asymptotic behaviour of problems as the number of variables becomes larger this paper is concerned with the location of the phase transition in finite problems the accuracy of a prediction based on the expected number of solutions is discussed it is shown that the variance of the number of solutions can be used to set bounds on the phase transition and to indicate the accuracy of the prediction a class of sparse problems for which the prediction is known to be inaccurate is considered in detail it is shown that for these problems the phase transition depends on the topology of the constraint graph as well as on the tightness of the constraints

local search with edge weighting and configuration checking heuristics for minimum vertex cover the minimum vertex cover mvc problem is a well known combinatorial optimization problem of great importance in theory and applications in recent years local search has been shown to be an effective and promising approach to solve hard problems such as mvc in this paper we introduce two new local search algorithms for mvc called ewls edge weighting local search and ewcc edge weighting configuration checking the first algorithm ewls is an iterated local search algorithm that works with a partial vertex cover and utilizes an edge weighting scheme which updates edge weights when getting stuck in local optima nevertheless ewls has an instance dependent parameter further we propose a strategy called configuration checking for handling the cycling problem in local search this is used in designing a more efficient algorithm that has no instance dependent parameters which is referred to as ewcc unlike previous vertex based heuristics the configuration checking strategy considers the induced subgraph configurations when selecting a vertex to add into the current candidate solution a detailed experimental study is carried out using the well known dimacs and bhoslib benchmarks the experimental results conclude that ewls and ewcc are largely competitive on dimacs benchmarks where they outperform other current best heuristic algorithms on most hard instances and dominate on the hard random bhoslib benchmarks moreover ewcc makes a significant improvement over ewls while both ewls and ewcc set a new record on a twenty year challenge instance further ewcc performs quite well even on structured instances in comparison to the best exact algorithm we know we also study the run time behavior of ewls and ewcc which shows interesting properties of both algorithms

a probabilistic study of generalized solution concepts in satisfiability testing and constraint programming we study the probabilistic behaviour of solutions of random instances of the boolean satisfiability sat and constraint satisfaction problems csps that generalize the standard notion of a satisfying assignment our analysis focuses on a special type of generalized solutions the 1 0 super solutions for random instances of k sat we establish the exact threshold of the phase transition of the solution probability for k 3 and give upper and lower bounds on the threshold of the phase transition for the case of k 4 for csps we derive an upper bound on the threshold of having a 1 0 super solution asymptotically with probability 1 and establish a condition for the expected number of super solutions to grow exponentially

strong fatou 1 points of blaschke products this paper shows that to every countable set m on the unit circle there corresponds a blaschke product whose set of strong fatou 1 points contains m it also shows that some blaschke products have an uncountable set of strong fatou 1 points

radial limits of interpolating blaschke products it is shown that if n is a sequence of distinct points on the unit circle then for a sequence an of points in the closed unit disk there exists an interpolating blaschke product b with b n an for all n if and only if an is bounded away from zero this complements results of cargo carroll colwell belna and piranian on prescribing radial limits for blaschke products

convergent relaxations of polynomial optimization problems with noncommuting variables we consider optimization problems with polynomial inequality constraints in noncommuting variables these noncommuting variables are viewed as bounded operators on a hilbert space whose dimension is not fixed and the associated polynomial inequalities as semidefinite positivity constraints such problems arise naturally in quantum theory and quantum information science to solve them we introduce a hierarchy of semidefinite programming relaxations which generates a monotone sequence of lower bounds that converges to the optimal solution we also introduce a criterion to detect whether the global optimum is reached at a given relaxation step and show how to extract a global optimizer from the solution of the corresponding semidefinite programming problem

constructing an asymptotic phase transition in random binary constraint satisfaction problems the standard models used to generate random binary constraint satisfaction problems are described at the problem sizes studied experimentally a phase transition is seen as the constraint tightness is varied however achlioptas et al showed that if the problem size number of variables increases while the remaining parameters are kept constant asymptotically almost all instances are unsatisfiable in this paper an alternative scheme for one of the standard models is proposed in which both the number of values in each variable s domain and the average degree of the constraint graph are increased with problem size it is shown that with this scheme there is asymptotically a range of values of the constraint tightness in which instances are trivially satis able with probability at least 0 5 and a range in which instances are almost all unsatisfiable hence there is a crossover point at some value of the constraint tightness between these two ranges this scheme is compared to a similar scheme due to xu and li

quantifying multipartite nonlocality the nonlocal correlations of multipartite entangled states can be reproduced by a classical model if sufficiently many parties join together or if sufficiently many parties broadcast their measurement inputs the maximal number m of groups and the minimal number k of broadcasting parties that allow for the reproduction of a given set of correlations quantify their multipartite nonlocal content we show how upper bounds on m and lower bounds on k can be computed from the violation of the mermin svetlichny inequalities while n partite greenberger horne zeilinger states violate these inequalities maximally we find that w states violate them only by a very small amount

an empirical study of phase transitions in binary constraint satisfaction problems abstract an empirical study of randomly generated binary constraint satisfaction problems reveals that for problems with a given number of variables domain size and connectivity there is a critical level of constraint tightness at which a phase transition occurs at the phase transition problems change from being soluble to insoluble and the difficulty of problems increases dramatically a theory developed by williams and hogg 44 and independently developed by smith 37 predicts where the hardest problems should occur it is shown that the theory is in close agreement with the empirical results except when constraint graphs are sparse

models for random constraint satisfaction problems we introduce a class of models for random constraint satisfaction problems this class includes and generalizes many previously studied models we characterize those models from our class which are asymptotically interesting in the sense that the limiting probability of satisfiability changes significantly as the number of constraints increases we also discuss models which exhibit a sharp threshold for satisfiability in the sense that the limiting probability jumps from 0 to 1 suddenly as the number of constraints increases

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

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

finite blaschke product interpolation on the closed unit disc abstract we show how to construct all finite blaschke product solutions and the minimal scaled blaschke product solution to the nevanlinna pick interpolation problem in the open unit disc by solving eigenvalue problems of the interpolation data based on a result of jones and ruscheweyh we note that there always exists a finite blaschke product of degree at most n 1 that maps n distinct points in the closed unit disc of which at least one is on the unit circle into n arbitrary points in the closed unit disc provided that the points inside the unit circle form a positive semi definite pick matrix of full rank finally we discuss a numerical limiting procedure

message passing for vertex covers constructing a minimal vertex cover of a graph can be seen as a prototype for a combinatorial optimization problem under hard constraints in this paper we develop and analyze message passing techniques namely warning and survey propagation which serve as efficient heuristic algorithms for solving these computational hard problems we show also how previously obtained results on the typical case behavior of vertex covers of random graphs can be recovered starting from the message passing equations and how they can be extended

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

threshold values stability analysis and high q asymptotics for the coloring problem on random graphs we consider the problem of coloring erdos renyi and regular random graphs of finite connectivity using q colors it has been studied so far using the cavity approach within the so called one step replica symmetry breaking 1rsb ansatz we derive a general criterion for the validity of this ansatz and applying it to the ground state we provide evidence that the 1rsb solution gives exact threshold values c q for the transition from the colorable to the uncolorable phase with q colors we also study the asymptotic thresholds for q 1 finding c q 2q ln q ln q 1 o 1 in perfect agreement with rigorous mathematical bounds as well as the nature of excited states and give a global phase diagram of the problem

a review on algorithms for maximum clique problems the maximum clique problem mcp is to determine in a graph a clique i e a complete subgraph of maximum cardinality the mcp is notable for its capability of modeling other combinatorial problems and real world applications as one of the most studied np hard problems many algorithms are available in the literature and new methods are continually being proposed given that the two existing surveys on the mcp date back to 1994 and 1999 respectively one primary goal of this paper is to provide an updated and comprehensive review on both exact and heuristic mcp algorithms with a special focus on recent developments to be informative we identify the general framework followed by these algorithms and pinpoint the key ingredients that make them successful by classifying the main search strategies and putting forward the critical elements of the most relevant clique methods this review intends to encourage future development of more powerful methods and motivate new applications of the clique approaches

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 network source location problem ground state energy entropy and effects of freezing ground state entropy of the network source location problem is evaluated at both the replica symmetric level and one step replica symmetry breaking level using the entropic cavity method the regime that is a focus of this study is closely related to the vertex cover problem with randomly quenched covered nodes the resulting entropic message passing inspired decimation and reinforcement algorithms are used to identify the optimal location of sources in single instances of transportation networks the conventional belief propagation without taking the entropic effect into account is also compared we find that in the glassy phase the entropic message passing inspired decimation yields a lower ground state energy compared to the belief propagation without taking the entropic effect using the extremal optimization algorithm we study the ground state energy and the fraction of frozen hubs and extend the algorithm to collect statistics of the entropy the theoretical results are compared with the extremal optimization results

bell s theorem for general n qubit states we derive a single general bell inequality which is a sufficient and necessary condition for the correlation function for n particles to be describable in a local and realistic picture for the case in which measurements on each particle can be chosen between two arbitrary dichotomic observables we also derive a necessary and sufficient condition for an arbitrary n qubit mixed state to violate this inequality this condition is a generalization and reformulation of the horodecki family condition for two qubits

device independent witnesses of genuine multipartite entanglement we consider the problem of determining whether genuine multipartite entanglement was produced in an experiment without relying on a characterization of the systems observed or of the measurements performed we present an n partite inequality that is satisfied by all correlations produced by measurements on biseparable quantum states but which can be violated by n partite entangled states such as greenberger horne zeilinger states in contrast to traditional entanglement witnesses the violation of this inequality implies that the state is not biseparable independently of the hilbert space dimension and of the measured operators violation of this inequality does not imply however genuine multipartite nonlocality we show more generically how the problem of identifying genuine tripartite entanglement in a deviceindependent way can be addressed through semidefinite programming

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

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

bell inequalities for arbitrarily high dimensional systems we develop a novel approach to bell inequalities based on a constraint that the correlations exhibited by local variable theories must satisfy this is used to construct a family of bell inequalities for bipartite quantum systems of arbitrarily high dimensionality which are strongly resistant to noise in particular our work gives an analytic description of previous numerical results and generalizes them to arbitrarily high dimensionality

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

family of bell like inequalities as device independent witnesses for entanglement depth we present a simple family of bell inequalities applicable to a scenario involving arbitrarily many parties each of which performs two binary outcome measurements we show that these inequalities are members of the complete set of full correlation bell inequalities discovered by werner wolf ifmmode dot z else z fi ukowski brukner for scenarios involving a small number of parties we further verify that these inequalities are facet defining for the convex set of bell local correlations moreover we show that the amount of quantum violation of these inequalities naturally manifests the extent to which the underlying system is genuinely many body entangled in other words our bell inequalities when supplemented with the appropriate quantum bounds naturally serve as device independent witnesses for entanglement depth allowing one to certify genuine k partite entanglement in an arbitrary n ensuremath ge k partite scenario without relying on any assumption about the measurements being performed or the dimension of the underlying physical system a brief comparison is made between our witnesses and those based on some other bell inequalities as well as quantum fisher information a family of witnesses for genuine k partite nonlocality applicable to an arbitrary n ensuremath ge k partite scenario based on our bell inequalities is also presented

ant colony algorithm with a strengthened negative feedback mechanism for constraint satisfaction problems describing the definition and usage of negative feedback in ant colony optimization proposing a new negative pheromone matrix and combined with positive pheromone matrix in probability formula a novel strengthening negative feedback mechanism is proposed to improve efficiency and avoid local optimal detail experiment design has been given for evaluating the proposed approaches based on different standard real datasets and synthetically generated datasets ant colony optimization aco is an efficient way to solve binary constraint satisfaction problems csps in recent years new improvements have only considered enhancing the positive feedback to increase the convergence speed however through the study and analysis of these enhanced aco algorithms we determined that they still suffer from the problem of easily getting in locally optimal solutions thus an improved aco algorithm with a strengthened negative feedback mechanism is designed to tackle csps this new algorithm takes advantage of search history information and continually obtains failure experience to guide the ant swarm exploring the unknown space during the optimization process the starting point of this algorithm is to utilize the negative feedback to improve the diversity of solutions finally we use 24 csp samples and 25 queen samples to perform experiments compare this algorithm with other related algorithms and conduct performance assessment the preliminary results show that aco with negative feedback outperforms the compared algorithms in identifying high quality solutions

the scaling window of the model d k csp abstract consider the model d k csp which consists of a set of n variables with the domain size d and a set of t r n ln d ln p constraints where each constraint binds k distinct variables and permits p d k tuples of values we prove that the exact phase transition of the model d k csp occurs at r c 1 as long as k ln d ln n where satisfies that 1 2 1 2 2 k 1 e 1 e 0 is a sufficiently small constant more importantly we establish the finite size scaling about this transition specifically let 0 be a given constant the probability of satisfiability of the model d k csp is greater than 1 if r 1 1 n 1 ln d is dependent on the control parameters and is less than if r 1 1 n ln d

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

many hard examples in exact phase transitions this paper analyzes the resolution complexity of two random constraint satisfaction problem csp models i e model rb rd for which we can establish the existence of phase transitions and identify the threshold points exactly by encoding csps into cnf formulas it is proved that almost all instances of model rb rd have no tree like resolution proofs of less than exponential size thus we not only introduce new families of csps and cnf formulas hard to solve which can be useful in the experimental evaluation of csp and sat algorithms but also propose models with both many hard instances and exact phase transitions finally conclusions are presented as well as a detailed comparison of model rb rd with the hamiltonian cycle problem and random 3 sat which respectively exhibit three different kinds of phase transition behavior in np complete problems