a polynomial time approximation algorithm for the permanent of a matrix with nonnegative entries we present a polynomial time randomized algorithm for estimating the permanent of an arbitrary n n matrix with nonnegative entries this algorithm technically a fully polynomial randomized approximation scheme computes an approximation that is with high probability within arbitrarily small specified relative error of the true value of the permanent

approximating bayesian belief networks by arc removal i propose a general framework for approximating bayesian belief networks through model simplification by arc removal given an upper bound on the absolute error allowed on the prior and posterior probability distributions of the approximated network a subset of arcs is removed thereby speeding up probabilistic inference

reconstruction of markov random fields from samples some observations and algorithms markov random fields are used to model high dimensional distributions in a number of applied areas much recent interest has been devoted to the reconstruction of the dependency structure from independent samples from the markov random fields we analyze a simple algorithm for reconstructing the underlying graph defining a markov random field on n nodes and maximum degree d given observations we show that under mild nondegeneracy conditions it reconstructs the generating graph with high probability using theta d epsilon 2 delta 4 log n samples where epsilon delta depend on the local interactions for most local interactions epsilon delta are of order exp o d our results are optimal as a function of n up to a multiplicative constant depending on d and the strength of the local interactions our results seem to be the first results for general models that guarantee that the generating model is reconstructed furthermore we provide explicit o n d 2 epsilon 2 delta

approximating maps for belief networks is np hard and other theorems abstract finding maximum a posteriori map assignments also called most probable explanations is an important problem on bayesian belief networks shimony has shown that finding maps is np hard in this paper we show that approximating maps with a constant ratio bound is also np hard in addition we examine the complexity of two related problems which have been mentioned in the literature we show that given the map for a belief network and evidence set or the family of maps if the optimal is not unique it remains np hard to find or approximate alternative next best explanations furthermore we show that given the map or maps for a belief network and an initial evidence set it is also np hard to find or approximate the map assignment for the same belief network with a modified evidence set that differs from the initial set by the addition or removal of even a single node assignment finally we show that our main result applies to networks with constrained in degree and out degree applies to randomized approximation and even still applies if the ratio bound instead of being constant is allowed to be a polynomial function of various aspects of the network topology

an improved fully polynomial randomized approximation scheme fpras for counting the number of hamiltonian cycles in dense digraphs we propose an improved algorithm for counting the number of hamiltonian cycles in a directed graph the basic idea of the method is sequential acceptance rejection which is successfully used in approximating the number of perfect matchings in dense bipartite graphs as a consequence a new ratio of the number of hamiltonian cycles to the number of 1 factors is proposed based on this ratio we prove that our algorithm runs in expected time of o n 8 5 for dense problems this improves the markov chain monte carlo method the most powerful existing method by a factor of at least n 4 5 logn 4 in running time this class of dense problems is shown to be nontrivial in counting in the sense that they are p complete

two approximate algorithms for model counting model counting is the problem of computing the number of models or satisfying assignments for a given propositional formula and is p complete owing to its inherent complexity approximate model counting is an alternative in practice model counting using the extension rule is an exact method and is considered as an alternative to resolution based methods for model counting based on the exact method we propose two approximate model counting algorithms and prove the time complexity of the algorithms experimental results show that they have good performance in the accuracy and efficiency

necessary conditions for vector valued operator inequalities in harmonic analysis via a random construction we establish necessary conditions for l p ell q inequalities for certain families of operators arising in harmonic analysis in particular we consider dilates of a convolution kernel with compactly supported fourier transform vector maximal functions acting on classes of entire functions of exponential type and a characterization of sobolev spaces by square functions and pointwise moduli of smoothness

counting and sampling minimum s t cuts in weighted planar graphs in polynomial time we give an o nd nlogn algorithm computing the number of minimum s t cuts in weighted planar graphs where n is the number of vertices and d is the length of the shortest s t path in the corresponding unweighted graph previously ball and provan gave a polynomial time algorithm for unweighted planar graphs with both s and t lying on the outer face our results hold for all locations of s and t and weighted graphs and have direct applications in computer vision

well posedness of the einstein euler system in asymptotically flat spacetimes the constraint equations this paper deals with the construction of initial data for the coupled einstein euler system we consider the condition where the energy density might vanish or tend to zero at infinity and where the pressure is a fractional power of the energy density in order to achieve our goals we use a type of weighted sobolev space of fractional order par the common lichnerowicz york scaling method choquet bruhat and york 1980 9 cantor 1979 7 for solving the constraint equations cannot be applied here directly the basic problem is that the matter sources are scaled conformally and the fluid variables have to be recovered from the conformally transformed matter sources this problem has been addressed although in a different context by dain and nagy 2002 11 we show that if the matter variables are restricted to a certain region then the einstein constraint equations have a unique solution in the weighted sobolev spaces of fractional order the regularity depends upon the fractional power of the equation of state

and or branch and bound search for combinatorial optimization in graphical models this is the first of two papers presenting and evaluating the power of a new framework for combinatorial optimization in graphical models based on and or search spaces we introduce a new generation of depth first branch and bound algorithms that explore the and or search tree using static and dynamic variable orderings the virtue of the and or representation of the search space is that its size may be far smaller than that of a traditional or representation which can translate into significant time savings for search algorithms the focus of this paper is on linear space search which explores the and or search tree in the second paper we explore memory intensive and or search algorithms in conjunction with the and or search space we investigate the power of the mini bucket heuristics in both static and dynamic setups we focus on two most common optimization problems in graphical models finding the most probable explanation in bayesian networks and solving weighted csps in extensive empirical evaluations we demonstrate that the new and or branch and bound approach improves considerably over the traditional or search strategy and show how various variable ordering schemes impact the performance of the and or search scheme

on the hardness of approximate reasoning abstract many ai problems when formalized reduce to evaluating the probability that a propositional expression is true in this paper we show that this problem is computationally intractable even in surprisingly restricted cases and even if we settle for an approximation to this probability we consider various methods used in approximate reasoning such as computing degree of belief and bayesian belief networks as well as reasoning techniques such as constraint satisfaction and knowledge compilation that use approximation to avoid computational difficulties and reduce them to model counting problems over a propositional domain we prove that counting satisfying assignments of propositional languages is intractable even for horn and monotone formulae and even when the size of clauses and number of occurrences of the variables are extremely limited this should be contrasted with the case of deductive reasoning where horn theories and theories with binary clauses are distinguished by the existence of linear time satisfiability algorithms what is even more surprising is that as we show even approximating the number of satisfying assignments i e approximating approximate reasoning is intractable for most of these restricted theories we also identify some restricted classes of propositional formulae for which efficient algorithms for counting satisfying assignments can be given

on the maximum satisfiability of random formulas say that a k cnf a formula is p satisfiable if there exists a truth assignment satisfying a fraction 1 2 k pp 2 k of its clauses note that every k cnf formula is 0 satisfiable let fk n m denote a random k cnf formula on n variables with m clauses for every k 2 and every r 0 we determine p and e k eo k2 k 2 such that with probability tending to 1 as n a random k cnf formula fk n rn is p satisfiable but not pp satisfiable

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

the cutoff phenomenon in finite markov chains abstract natural mixing processes modeled by markov chains often show a sharp cutoff in their convergence to long time behavior this paper presents problems where the cutoff can be proved card shuffling the ehrenfests urn it shows that chains with polynomial growth drunkard s walk do not show cutoffs the best general understanding of such cutoffs high multiplicity of second eigenvalues due to symmetry is explored examples are given where the symmetry is broken but the cutoff phenomenon persists

approximating probabilistic inference in bayesian belief networks a belief network comprises a graphical representation of dependencies between variables of a domain and a set of conditional probabilities associated with each dependency unless rho np an efficient exact algorithm does not exist to compute probabilistic inference in belief networks stochastic simulation methods which often improve run times provide an alternative to exact inference algorithms such a stochastic simulation algorithm d bnras which is a randomized approximation scheme is presented to analyze the run time belief networks are parameterized by the dependence value d sub xi which is a measure of the cumulative strengths of the belief network dependencies given background evidence xi this parameterization defines the class of f dependence networks the run time of d bnras is polynomial when f is a polynomial function thus the results prove the existence of a class of belief networks for which inference approximation is polynomial and hence provably faster than any exact algorithm

random walks that find perfect objects and the lovasz local lemma we give an algorithmic local lemma by establishing a sufficient condition for the uniform random walk on a directed graph to reach a sink quickly our work is inspired by moser s entropic method proof of the lovasz local lemma lll for satisfiability and completely bypasses the probabilistic method formulation of the lll in particular our method works when the underlying state space is entirely unstructured similarly to moser s argument the key point is that the inevitability of reaching a sink is established by bounding the entropy of the walk as a function of time

improving variational methods via pairwise linear response identities inference methods are often formulated as variational approximations these approximations allow easy evaluation of statistics by marginalization or linear response but these estimates can be inconsistent we show that by introducing constraints on covariance one can ensure consistency of linear response with the variational parameters and in so doing inference of marginal probability distributions is improved for the bethe approximation and its generalizations improvements are achieved with simple choices of the constraints the approximations are presented as variational frameworks iterative procedures related to message passing are provided for finding the minima

ising models on power law random graphs we study a ferromagnetic ising model on random graphs with a power law degree distribution and compute the thermodynamic limit of the pressure when the mean degree is finite degree exponent 2 for which the random graph has a tree like structure for this we closely follow the analysis by dembo and montanari ann appl probab 20 2 565 592 2010 which assumes finite variance degrees 3 adapting it when necessary and also simplifying it when possible our results also apply in cases where the degree distribution does not obey a power law

approximate counting uniform generation and rapidly mixing markov chains the paper studies effective approximate solutions to combinatorial counting and uniform generation problems using a technique based on the simulation of ergodic markov chains it is shown that for self reducible structures almost uniform generation is possible in polynomial time provided only that randomised approximate counting to within some arbitrary polynomial factor is possible in polynomial time it follows that for self reducible structures polynomial time randomised algorithms for counting to within factors of the form 1 n are available either for all fl e r or for no fi e r a substantial part of the paper is devoted to investigating the rate of convergence of finite ergodic markov chains and a simple but powerful characterisation of rapid convergence for a broad class of chains based on a structural property of the underlying graph is established finally the general techniques of the paper are used to derive an almost uniform generation procedure for labelled graphs with a given degree sequence which is valid over a much wider range of degrees than previous methods this in turn leads to randomised approximate counting algorithms for these graphs with very good

adaptive simulated annealing a near optimal connection between sampling and counting we present a near optimal reduction from approximately counting the cardinality of a discrete set to approximately sampling elements of the set an important application of our work is to approximating the partition function z of a discrete system such as the ising model matchings or colorings of a graph the typical approach to estimating the partition function z at some desired inverse temperature is to define a sequence which we call a cooling schedule 0 e 0 for well studied problems such as estimating the partition function of the ising model or approximating the number of colorings or matchings of a graph our cooling schedule is of length o s n which implies an overall savings of o n in the running time of the approximate counting algorithm since roughly e samples are needed to estimate each ratio a similar improvement in the length of the cooling schedule was recently obtained by lovasz and vempala in the context of estimating the volume of convex bodies while our reduction is inspired by theirs the discrete analogue of their result turns out to be significantly more difficult whereas a fixed schedule suffices in their setting we prove that in the discrete setting we need an adaptive schedule that is the schedule depends on z more precisely we prove any nonadaptive cooling schedule has length at least o ln a and we present an algorithm to find an adaptive schedule of length o s ln a

the two possible values of the chromatic number of a random graph given d 0 let kd be the smallest integer k such that d 2k log k we prove that the chromatic number of a random graph g n d n is either kd or kd 1 almost surely

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

counting and sampling h colourings for counting problems in p which are essentially self reducible it is known that sampling and approximate counting are equivalent however many problems of interest do not have such a structure and there is already some evidence that this equivalence does not hold for the whole of p an intriguing example is the class of h colouring problems which have recently been the subject of much study and their natural generalisation to vertex and edge weighted versions particular cases of the counting to sampling reduction have been observed but it has been an open question as to how far these reductions might extend to any h and a general graph g here we give the first completely general counting to sampling reduction for every fixed h we show that the problem of approximately determining the partition function of weighted h colourings can be reduced to the problem of sampling these colourings from an approximately correct distribution in particular any rapidly mixing markov chain for sampling h colourings can be turned into an fpras for counting h colourings

counting by quantum eigenvalue estimation for every computation there corresponds the physical task of manipulating a starting state into an output state with a desired property as the classical theory of physics has been replaced by quantum physics it is interesting to consider the capabilities of a computer that can exploit the distinctive quantum features of nature the extra capabilities seem enormous for example with only an expected o square rootn evaluations of a function f 0 1 n 1 0 1 we can find a solution to f x 1 provided one exists another example is the ability to 1nd efficiently the order of an element g in a group by using a quantum computer to estimate a random eigenvalue of the unitary operator that multiplies by g in the group by using this eigenvalue estimation algorithm to estimate an eigenvalue of the unitary operator used in quantum searching we can approximately count the number of solutions to f x 1 this paper describes this eigenvector approach to quantum counting and related algorithms

a randomized algorithm to optimize over certain convex sets this paper presents a randomized polynomial time algorithm to nearly minimize a linear function over an up monotone convex set in the positive orthant given only by a membership oracle our original motivation for this is a stochastic optimization problem called the component commonality problem in the literature

lee yang theorems and the complexity of computing averages we study the complexity of computing average quantities related to spin systems such as the mean magnetization and susceptibility in the ferromagnetic ising model and the average dimer count or average size of a matching in the monomer dimer model by establishing connections between the complexity of computing these averages and the location of the complex zeros of the partition function we show that these averages are p hard to compute and hence under standard assumptions computationally intractable in the case of the ising model our approach requires us to prove an extension of the famous lee yang theorem from the 1950s

probabilistic data exchange the work reported here lays the foundations of data exchange in the presence of probabilistic data this requires rethinking the very basic concepts of traditional data exchange such as solution universal solution and the certain answers of target queries we develop a framework for data exchange over probabilistic databases and make a case for its coherence and robustness this framework applies to arbitrary schema mappings and finite or countably infinite probability spaces on the source and target instances after establishing this framework and formulating the key concepts we study the application of the framework to a concrete and practical setting where probabilistic databases are compactly encoded by means of annotations formulated over random boolean variables in this setting we study the problems of testing for the existence of solutions and universal solutions materializing such solutions and evaluating target queries for unions of conjunctive queries in both the exact sense and the approximate sense for each of the problems we carry out a complexity analysis based on properties of the annotation for various classes of dependencies finally we show that the framework and results easily and completely generalize to allow not only the data but also the schema mapping itself to be probabilistic

new constructive aspects of the lovasz local lemma the lovasz local lemma lll is a powerful tool that gives sufficient conditions for avoiding all of a given set of bad events with positive probability a series of results have provided algorithms to efficiently construct structures whose existence is non constructively guaranteed by the lll culminating in the recent breakthrough of moser and tardos 2010 for the full asymmetric lll we show that the output distribution of the moser tardos algorithm well approximates the conditional lll distribution the distribution obtained by conditioning on all bad events being avoided we show how a known bound on the probabilities of events in this distribution can be used for further probabilistic analysis and give new constructive and nonconstructive results we also show that when a lll application provides a small amount of slack the number of resamplings of the moser tardos algorithm is nearly linear in the number of underlying independent variables not events and can thus be used to give efficient constructions in cases where the underlying proof applies the lll to super polynomially many events even in cases where finding a bad event that holds is computationally hard we show that applying the algorithm to avoid a polynomial sized core subset of bad events leads to a desired outcome with high probability this is shown via a simple union bound over the probabilities of non core events in the conditional lll distribution and automatically leads to simple and efficient monte carlo and in most cases rnc algorithms we demonstrate this idea on several applications we give the first constant factor approximation algorithm for the santa claus problem by making a lll based proof of feige constructive we provide monte carlo algorithms for acyclic edge coloring nonrepetitive graph colorings and ramsey type graphs in all these applications the algorithm falls directly out of the non constructive lll based proof our algorithms are very simple often provide better bounds than previous algorithms and are in several cases the first efficient algorithms known as a second type of application we show that the properties of the conditional lll distribution can be used in cases beyond the critical dependency threshold of the lll avoiding all bad events is impossible in these cases as the first even nonconstructive result of this kind we show that by sampling a selected smaller core from the lll distribution we can avoid a fraction of bad events that is higher than the expectation max k sat is an illustrative example of this

finding maps for belief networks is np hard given a probabilistic world model an important problem is to find the maximum a posteriori probability map instantiation of all the random variables given the evidence numerous researchers using such models employ some graph representation for the distributions such as a bayesian belief network this representation simplifies the complexity of specifying the distributions from exponential in n the number of variables in the model to linear in n in many interesting cases we show however that finding the map is np hard in the general case when these representations are used even if the size of the representation happens to be linear in n furthermore minor modifications to the proof show that the problem remains np hard for various restrictions of the topology of the graphs the same technique can be applied to the results of a related paper by cooper to further restrict belief network topology in the proof that probabilistic inference is np hard

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

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

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

probabilistic conflicts in a search algorithm for estimating posterior probabilities in bayesian networks abstract this paper presents a search algorithm for estimating posterior probabilities in discrete bayesian networks it shows how conflicts as used in consistency based diagnosis can be adapted to speed up the search this algorithm is especially suited to the case where there are skewed distributions although nothing about the algorithm or the definitions depends on skewness of distributions the general idea is to forward simulate the network based on the normal values for each variable the value with high probability given its parents when a predicted value is at odds with the observations we analyse which variables were responsible for the expectation failure these form a conflict and continue forward simulation considering different values for these variables this results in a set of possible worlds from which posterior probabilities together with error bounds can be derived empirical results with bayesian networks having tens of thousands of nodes are presented

rough solutions of the einstein vacuum equations this is the first in a series of papers in which we initiate the study of very rough solutions to the initial value problem for the einstein vacuum equations expressed relative to wave coordinates by very rough we mean solutions which cannot be constructed by the classical techniques of energy estimates and sobolev inequalities following kl ro we develop new analytic methods based on strichartz type inequalities which result in a gain of half a derivative relative to the classical result our methods blend paradifferential techniques with a geometric approach to the derivation of decay estimates the latter allows us to take full advantage of the specific structure of the einstein equations

on the number of occurrences of a symbol in words of regular languages we study the random variable yn representing the number of occurrences of a symbol a in a word of length n chosen at random in a regular language l a b where the random choice is defined via a non negative rational formal series r of support l assuming that the transition matrix associated with r is primitive we obtain asymptotic estimates for the mean value and the variance of yn and present a central limit theorem for its distribution under a further condition on such a matrix we also derive an asymptotic approximation of the discrete fourier transform of yn that allows to prove a local limit theorem for yn further consequences of our analysis concern the growth of the coefficients in rational formal series in particular it turns out that for a wide class of regular languages l the maximum number of words of length n in l having the same number of occurrences of a given symbol is of the order of growth n n for some constant 1

the weak limit of ising models on locally tree like graphs we consider the ising model with inverse temperature and without external field on sequences of graphs g n which converge locally to the k regular tree we show that for such graphs the ising measure locally weakly converges to the symmetric mixture of the ising model with boundary conditions and the boundary conditions on the k regular tree with inverse temperature in the case where the graphs g n are expanders we derive a more detailed understanding by showing convergence of the ising measure conditional on positive magnetization sum of spins to the measure on the tree

factor models on locally tree like graphs we consider homogeneous factor models on uniformly sparse graph sequences converging locally to a unimodular random tree t and study the existence of the free energy density the limit of the log partition function divided by the number of vertices n as n tends to infinity we provide a new interpolation scheme and use it to prove existence of and to explicitly compute the quantity subject to uniqueness of a relevant gibbs measure for the factor model on t by way of example we compute for the independent set or hard core model at low fugacity for the ferromagnetic ising model at all parameter values and for the ferromagnetic potts model with both weak enough and strong enough interactions even beyond uniqueness regimes our interpolation provides useful explicit bounds on in the regimes in which we establish existence of the limit we show that it coincides with the bethe free energy functional evaluated at a suitable fixed point of the belief propagation bethe recursions on t in the special case that t has a galton watson law this formula coincides with the nonrigorous bethe prediction obtained by statistical physicists using the replica or cavity methods thus our work is a rigorous generalization of these heuristic calculations to the broader class of sparse graph sequences converging locally to trees we also provide a variational characterization for the bethe prediction in this general setting which is of independent interest

improved mixing condition on the grid for counting and sampling independent sets the hard core model has received much attention in the past couple of decades as a lattice gas model with hard constraints in statistical physics a multicast model of calls in communication networks and as a weighted independent set problem in combinatorics probability and theoretical computer science in this model each independent set i in a graph g is weighted proportionally to i for a positive real parameter for large computing the partition function namely the normalizing constant which makes the weighting a probability distribution on a finite graph on graphs of maximum degree 3 is a well known computationally challenging problem more concretely let lambda c mathbb t delta denote the critical value for the so called uniqueness threshold of the hard core model on the infinite regular tree recent breakthrough results of weitz proceedings of the 38th annual acm symposium on theory of computing stoc pp 140 149 2006 and sly proceedings of the 51st annual ieee symposium on foundations of computer science focs pp 287 296 2010 have identified lambda c mathbb t delta as a threshold where the hardness of estimating the above partition function undergoes a computational transition we focus on the well studied particular case of the square lattice mathbb z 2 and provide a new lower bound for the uniqueness threshold in particular taking it well above lambda c mathbb t 4 our technique refines and builds on the tree of self avoiding walks approach of weitz resulting in a new technical sufficient criterion of wider applicability for establishing strong spatial mixing and hence uniqueness for the hard core model our new criterion achieves better bounds on strong spatial mixing when the graph has extra structure improving upon what can be achieved by just using the maximum degree applying our technique to mathbb z 2 we prove that strong spatial mixing holds for all 2 3882 improving upon the work of weitz that held for 27 16 1 6875 our results imply a fully polynomial deterministic approximation algorithm for estimating the partition function as well as rapid mixing of the associated glauber dynamics to sample from the hard core distribution

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

the complexity of approximating maps for belief networks with bounded probabilities abstract probabilistic inference and maximum a posteriori map explanation are two important and related problems on bayesian belief networks both problems are known to be np hard for both approximation and exact solution in 1997 dagum and luby showed that efficiently approximating probabilistic inference is possible for belief networks in which all probabilities are bounded away from 0 in this paper we show that the corresponding result for map explanation does not hold finding or approximating maps for belief networks remains np hard for belief networks with probabilities bounded within the range l u for any 0 l our results cover both deterministic and randomized approximation

local existence of solutions of self gravitating relativistic perfect fluids this paper deals with the evolution of the einstein gravitational fields which are coupled to a perfect fluid we consider the einstein euler system in asymptotically flat spacestimes and therefore use the condition that the energy density might vanish or tend to zero at infinity and that the pressure is a fractional power of the energy density in this setting we prove local in time existence uniqueness and well posedness of classical solutions the zero order term of our system contains an expression which might not be a c function and therefore causes an additional technical difficulty in order to achieve our goals we use a certain type of weighted sobolev space of fractional order in brauer and karp j diff eqs 251 1428 1446 2011 we constructed an initial data set for these of systems in the same type of weighted sobolev spaces

the complexity of approximating bounded degree boolean csp the degree of a csp instance is the maximum number of times that any variable appears in the scopes of constraints we consider the approximate counting problem for boolean csp with bounded degree instances for constraint languages containing the two unary constant relations 0 and 1 when the maximum allowed degree is large enough at least 6 we obtain a complete classification of the complexity of this problem it is exactly solvable in polynomial time if every relation in the constraint language is affine it is equivalent to the problem of approximately counting independent sets in bipartite graphs if every relation can be expressed as conjunctions of 0 1 and binary implication otherwise there is no fpras unless np rp for lower degree bounds additional cases arise where the complexity is related to the complexity of approximately counting independent sets in hypergraphs

the computational complexity of probabilistic inference using bayesian belief networks research note abstract bayesian belief networks provide a natural efficient method for representing probabilistic dependencies among a set of variables for these reasons numerous researchers are exploring the use of belief networks as a knowledge representation in artificial intelligence algorithms have been developed previously for efficient probabilistic inference using special classes of belief networks more general classes of belief networks however have eluded efforts to develop efficient inference algorithms we show that probabilistic inference using belief networks is np hard therefore it seems unlikely that an exact algorithm can be developed to perform probabilistic inference efficiently over all classes of belief networks this result suggests that research should be directed away from the search for a general efficient probabilistic inference algorithm and toward the design of efficient special case average case and approximation algorithms

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

diagnosis under compound effects and multiple causes by means of the conditional causal possibility approach abstract the paper addresses uncertain reasoning on a causal model given by two layered networks where nodes in one layer express possible causes and those in the other are possible effects uncertainty of causalities is expressed in a novel manner i e by conditional causal possibilities the expression has two advantages over the conventional way with conditional possibilities it expresses the exact degrees of possibility of causalities and the number of necessary conditional causal possibilities is far smaller than that of conditional possibilities however it has a weakness that it cannot handle causalities with compound effects such as synergistic and canceling effects by multiple causes the paper discusses the weakness and proposes a solution first it discusses how to deal with the compound effects and proposes a new causal model with conditional causal possibilities by multiple causes then it defines a causality consistency problem that calculates possibility of a hypothesis given some observed events and shows a way to solve the problem

fluctuation results for general block spin ising models we study a block spin mean field ising model i e a model of spins in which the vertices are divided into a finite number of blocks with each block having a fixed proportion of vertices and where pair interactions are given according to their blocks for the vector of block magnetizations we prove large deviation principles and central limit theorems under general assumptions for the block interaction matrix using the exchangeable pair approach of stein s method we establish a rate of convergence in the central limit theorem for the block magnetization vector in the high temperature regime

dynamic uncertain causality graph for knowledge representation and probabilistic reasoning statistics base matrix and application graphical models for probabilistic reasoning are now in widespread use many approaches have been developed such as bayesian network a newly developed approach named as dynamic uncertain causality graph ducg is initially presented in a previous paper in which only the inference algorithm in terms of individual events and probabilities is addressed in this paper we first explain the statistic basis of ducg then we extend the algorithm to the form of matrices of events and probabilities it is revealed that the representation of ducg can be incomplete and the exact probabilistic inference may still be made a real application of ducg for fault diagnoses of a generator system of a nuclear power plant is demonstrated which involves variables most inferences take with a laptop computer the causal logic between inference result and observations is graphically displayed to users so that they know not only the result but also why the result obtained

approximation algorithms for two state anti ferromagnetic spin systems on bounded degree graphs we show that for the anti ferromagnetic ising model on the bethe lattice weak spatial mixing implies strong spatial mixing as a by product of our analysis we obtain what is to the best of our knowledge the first rigorous proof of the uniqueness threshold for the anti ferromagnetic ising model with non zero external field on the bethe lattice following a method due to weitz 15 we then use the equivalence between weak and strong spatial mixing to give a deterministic fully polynomial time approximation scheme for the partition function of the anti ferromagnetic ising model with arbitrary field on graphs of degree at most d throughout the uniqueness region of the gibbs measure on the infinite d regular tree by a standard correspondence our results translate to arbitrary two state anti ferromagnetic spin systems with soft constraints subsequent to a preliminary version of this paper sly and sun 13 have shown that our results are optimal in the sense that under standard complexity theoretic assumptions there does not exist a fully polynomial time approximation scheme for the partition function of such spin systems on graphs of maximum degree d for parameters outside the uniqueness region taken together the results of 13 and of this paper therefore indicate a tight relationship between complexity theory and phase transition phenomena in two state anti ferromagnetic spin systems

a random polynomial time algorithm for approximating the volume of convex bodies a randomized polynomial time algorithm for approximating the volume of a convex body k in n dimensional euclidean space is presented the proof of correctness of the algorithm relies on recent theory of rapidly mixing markov chains and isoperimetric inequalities to show that a certain random walk can be used to sample nearly uniformly from within k

on the boundedness of pseudo differential operators on triebel lizorkin and besov spaces abstract in this work we show endpoint boundedness properties of pseudo differential operators of type 0 1 on triebel lizorkin and besov spaces our results are sharp and they also cover operators defined by compound symbols

possibilistic causality consistency problem based on asymmetrically valued causal model the paper addresses uncertain reasoning based on causal knowledge given by two layered networks where nodes in one layer express possible causes and those in the other are possible effects uncertainties of the causalities are given by conditional causal possibilities which were proposed to express the exact degrees of possibility of causalities the expression of the uncertainty also has an advantage over the conventional conditional possibilities in the number of possibilistic values that should be given as a priori knowledge the number of conditional causal possibilities given as knowledge is far smaller than that of conventional conditional possibilities the paper starts with the definition of a causal model called asymmetrically valued causal model the conditional causal possibilities are defined on the causal model and their mathematical properties are discussed then the paper defines the possibilistic causality consistency problem based on the proposed model and shows how to solve the problem the discussed problem is one to calculate the conditional possibility of a hypothesis about presence and absence of unknown events when the states of some other events are known

approximate belief updating in max 2 connected bayes networks is np hard a max 2 connected bayes network is one where there are at most 2 distinct directed paths between any two nodes we show that even for this restricted topology null evidence belief updating is hard to approximate

mini buckets a general scheme for bounded inference this article presents a class of approximation algorithms that extend the idea of bounded complexity inference inspired by successful constraint propagation algorithms to probabilistic inference and combinatorial optimization the idea is to bound the dimensionality of dependencies created by inference algorithms this yields a parameterized scheme called mini buckets that offers adjustable trade off between accuracy and efficiency the mini bucket approach to optimization problems such as finding the most probable explanation mpe in bayesian networks generates both an approximate solution and bounds on the solution quality we present empirical results demonstrating successful performance of the proposed approximation scheme for the mpe task both on randomly generated problems and on realistic domains such as medical diagnosis and probabilistic decoding

local existence of solutions to the euler poisson system including densities without compact support abstract local existence and well posedness for a class of solutions for the euler poisson system is shown these solutions have a density which either falls off at infinity or has compact support the solutions have finite mass finite energy functional and include the static spherical solutions for 6 5 the result is achieved by using weighted sobolev spaces of fractional order and a new non linear estimate which allows to estimate the physical density by the regularised non linear matter variable gamblin also has studied this setting but using very different functional spaces however we believe that the functional setting we use is more appropriate to describe a physical isolated body and more suitable to study the newtonian limit

the replica symmetric solution for potts models on d regular graphs we establish an explicit formula for the limiting free energy density log partition function divided by the number of vertices for ferromagnetic potts models on uniformly sparse graph sequences converging locally to the d regular tree for d even covering all temperature regimes this formula coincides with the bethe free energy functional evaluated at a suitable fixed point of the belief propagation recursion on the d regular tree the so called replica symmetric solution for uniformly random d regular graphs we further show that the replica symmetric bethe formula is an upper bound for the asymptotic free energy for any model with permissive interactions

approximate counting the lovasz local lemma and inference in graphical models in this article we introduce a new approach to approximate counting in bounded degree systems with higher order constraints our main result is an algorithm to approximately count the number of solutions to a cnf formula when the width is logarithmic in the maximum degree this closes an exponential gap between the known upper and lower bounds moreover our algorithm extends straightforwardly to approximate sampling which shows that under lovasz local lemma like conditions it is not only possible to find a satisfying assignment it is also possible to generate one approximately uniformly at random from the set of all satisfying assignments our approach is a significant departure from earlier techniques in approximate counting and is based on a framework to bootstrap an oracle for computing marginal probabilities on individual variables finally we give an application of our results to show that it is algorithmically possible to sample from the posterior distribution in an interesting class of graphical models

fourier multiplier theorems for triebel lizorkin spaces in this paper we study sharp generalizations of dot f p 0 q multiplier theorem of mikhlin hormander type the class of multipliers that we consider involves herz spaces k u s t plancherel s theorem proves widehat l s 2 k 2 s 2 and we study the optimal triple u t s for which sup k in mathbb z vert m 2 k cdot varphi vee vert k u s t infty implies dot f p 0 q boundedness of multiplier operator t m where varphi is a cutoff function our result also covers the bmo type space dot f infty 0 q

approximating probabilistic inference in bayesian belief networks is np hard abstract it is known that exact computation of conditional probabilities in belief networks is np hard many investigators in the ai community have tacitly assumed that algorithms for performing approximate inference with belief networks are of polynomial complexity indeed special cases of approximate inference can be performed in time polynomial in the input size however we have discovered that the general problem of approximating conditional probabilities with belief networks like exact inference resides in the np hard complexity class we develop a complexity analysis to elucidate the difficulty of approximate probabilistic inference more specifically we show that the existence of a polynomial time relative approximation algorithm for major classes of problem instances implies that np p we present our proof and explore the implications of the result

the newtonian limit for perfect fluids we prove that there exists a class of non stationary solutions to the einstein euler equations which have a newtonian limit the proof of this result is based on a symmetric hyperbolic formulation of the einstein euler equations which contains a singular parameter vt c where vt is a characteristic velocity scale associated with the fluid and c is the speed of light the symmetric hyperbolic formulation allows us to derive e independent energy estimates on weighted sobolev spaces these estimates are the main tool used to analyze the behavior of solutions in the limit 0

an optimal approximation algorithm for bayesian inference abstract approximating the inference probability pr x x e e in any sense even for a single evidence node e is np hard this result holds for belief networks that are allowed to contain extreme conditional probabilities that is conditional probabilities arbitrarily close to 0 nevertheless all previous approximation algorithms have failed to approximate efficiently many inferences even for belief networks without extreme conditional probabilities we prove that we can approximate efficiently probabilistic inference in belief networks without extreme conditional probabilities we construct a randomized approximation algorithm the bounded variance algorithm that is a variant of the known likelihood weighting algorithm the bounded variance algorithm is the first algorithm with provably fast inference approximation on all belief networks without extreme conditional probabilities from the bounded variance algorithm we construct a deterministic approximation algorithm using current advances in the theory of pseudorandom generators in contrast to the exponential worst case behavior of all previous deterministic approximations the deterministic bounded variance algorithm approximates inference probabilities in worst case time that is subexponential 2 log n d for some integer d that is a linear function of the depth of the belief network

the expressibility of functions on the boolean domain with applications to counting csps an important tool in the study of the complexity of constraint satisfaction problems csps is the notion of a relational clone which is the set of all relations expressible using primitive positive formulas over a particular set of base relations post s lattice gives a complete classification of all boolean relational clones and this has been used to classify the computational difficulty of csps motivated by a desire to understand the computational complexity of weighted counting csps we develop an analogous notion of functional clones and study the landscape of these clones one of these clones is the collection of log supermodular lsm functions which turns out to play a significant role in classifying counting csps in the conservative case where all nonnegative unary functions are available we show that there are no functional clones lying strictly between the clone of lsm functions and the total clone containing all functions thus any counting csp that contains a single nontrivial non lsm function is computationally as hard to approximate as any problem in p furthermore we show that any nontrivial functional clone in a sense that will be made precise contains the binary function implies as a consequence in the conservative case all nontrivial counting csps are as hard to approximate as bis the problem of counting independent sets in a bipartite graph given the complexity theoretic results it is natural to ask whether the implies clone is equivalent to the clone of lsm functions we use the mobius transform and the fourier transform to show that these clones coincide precisely up to arity 3 it is an intriguing open question whether the lsm clone is finitely generated finally we investigate functional clones in which only restricted classes of unary functions are available

multi group binary choice with social interaction and a random communication structure a random graph approach abstract we construct and analyze a random graph model for discrete choice with social interaction and several groups of equal size we concentrate on the case of two groups of equal sizes and we allow the interaction strength within a group to differ from the interaction strength between the two groups given that the resulting graph is sufficiently dense we show that with probability 1 the average decision in each of the two groups is the same as in the fully connected model in particular we show that there is a phase transition if the interaction among a group and between the groups is strong enough the average decision per group will either be positive or negative and the decision of the two groups will be correlated we also compute the free energy per particle in our model

a discrete transform and decompositions of distribution spaces abstract we study a representation formula of the form q q q for a distribution on r n this formula is obtained by discretizing and localizing a standard littlewood paley decomposition the map taking to the sequence q q with q running over the dyadic cubes in r n is called the transform the functions q and q have a particularly simple form moreover most of the familiar distribution spaces lp spaces 1 r n we also consider pointwise multipliers for the characteristic function of a domain we obtain boundedness results for a general class of domains which properly includes lipschitz domains several interpolation methods are easily analyzed via the sequence spaces for real interpolation we obtain among other things an extension to the case p 0 this in turn gives a new approach to the traditional atomic decomposition of hardy spaces

inapproximability of the tutte polynomial the tutte polynomial of a graph g is a two variable polynomial t g x y that encodes many interesting properties of the graph we study the complexity of the following problem for rationals x and y take as input a graph g and output a value which is a good approximation to t g x y jaeger et al have completely mapped the complexity of exactly computing the tutte polynomial they have shown that this is p hard except along the hyperbola x 1 y 1 1 and at four special points we are interested in determining for which points x y there is a fully polynomial randomised approximation scheme fpras for t g x y under the assumption rp np we prove that there is no fpras at x y if x y is in one of the half planes x np there is no fpras at the point x y 0 email protected when l 2 is a positive integer thus there is no fpras for counting nowhere zero l flows for l 2 this is an interesting consequence of our work since the corresponding decision problem is in p for example for l 6 although our main concern is to distinguish regions of the tutte plane that admit an fpras from those that do not we also note that the latter regions exhibit different levels of intractability at certain points x y for example the integer points on the x axis or any point in the positive quadrant there is a randomised approximation scheme for t g x y that runs in polynomial time using an oracle for an np predicate on the other hand we identify a region of points x y at which even approximating t g x y is as hard as p

approximating the number of double cut and join scenarios the huge number of solutions in genome rearrangement problems calls for algorithms for counting and sampling in the space of solutions rather than drawing one arbitrary scenario a closed formula exists for counting the number of dcj scenarios between co tailed genomes but no polynomial result has been published so far for arbitrary genomes we prove here that it admits a fully polynomial time randomized approximation scheme we use an mcmc almost uniform sampler and prove that it converges to the uniform distribution in fully polynomial time the mcmc can be used to quickly draw a sample of dcj scenarios from a prescribed distribution and test some hypotheses on genome evolution

complex interpolation and fourier multipliers for the spacesb p q s andf p q s of besov hardy sobolev type the case 0 p 0 q

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

fisher zeros and correlation decay in the ising model we study the complex zeros of the partition function of the ising model viewed as a polynomial in the interaction parameter these are known as fisher zeros in light of their introduction by fisher in 1965 fisher m e the nature of critical points in lecture notes in theoretical physics edited by brittin w e university of colorado press 1965 vol 7c while the zeros of the partition function as a polynomial in the field parameter have been extensively studied since the classical work of lee and yang comparatively little is known about fisher zeros for general graphs our main result shows that the zero field ising model has no fisher zeros in a complex neighborhood of the entire region of parameters where the model exhibits correlation decay in addition to shedding light on fisher zeros themselves this result also establishes a formal connection between two distinct notions of phase transition for the ising model the absence of complex zeros analyticity of the free energy density or the normalized logarithm of the partition function and decay of correlations with distance we also discuss the consequences of our result for efficient deterministic approximation of the partition function our proof relies heavily on algorithmic techniques notably weitz s self avoiding walk tree and as such belongs to a growing body of work that uses algorithmic methods to resolve classical questions in statistical physics we study the complex zeros of the partition function of the ising model viewed as a polynomial in the interaction parameter these are known as fisher zeros in light of their introduction by fisher in 1965 fisher m e the nature of critical points in lecture notes in theoretical physics edited by brittin w e university of colorado press 1965 vol 7c while the zeros of the partition function as a polynomial in the field parameter have been extensively studied since the classical work of lee and yang comparatively little is known about fisher zeros for general graphs our main result shows that the zero field ising model has no fisher zeros in a complex neighborhood of the entire region of parameters where the model exhibits correlation decay in addition to shedding light on fisher zeros themselves this result also establishes a formal connection between two distinct notions of phase transition for the ising model the absence of complex zeros analyticity of the free energy density

on the complexity of inference about probabilistic relational models abstract we investigate the complexity of probabilistic inference from knowledge bases that encode probability distributions on finite domain relational structures our interest here lies in the complexity in terms of the domain under consideration in a specific application instance we obtain the result that assuming netime etime this problem is not polynomial for reasonably expressive representation systems the main consequence of this result is that it is unlikely to find inference techniques with a better worst case behavior than the commonly employed strategy of constructing standard bayesian networks over ground atoms knowledge based model construction

counting hypergraph matchings up to uniqueness threshold abstract we study the problem of approximately counting matchings in hypergraphs of bounded maximum degree and maximum size of hyperedges with an activity parameter each matching m is assigned a weight m the counting problem is formulated as computing a partition function that gives the sum of the weights of all matchings in a hypergraph this problem unifies two extensively studied statistical physics models in approximate counting the hardcore model graph independent sets and the monomer dimer model graph matchings for this problem the critical activity c d d k d 1 d 1 is the threshold for the uniqueness of gibbs measures on the infinite d 1 uniform k 1 regular hypertree consider hypergraphs of maximum degree at most k 1 and maximum size of hyperedges at most d 1 we show that when c there is an fptas for computing the partition function and when c there is a ptas for computing the log partition function these algorithms are based on the decay of correlation strong spatial mixing property of gibbs distributions when 2 c there is no pras for the partition function or the log partition function unless np rp towards obtaining a sharp transition of computational complexity of approximate counting we study the local convergence from a sequence of finite hypergraphs to the infinite lattice with specified symmetry we show a surprising connection between the local convergence and the reversibility of a natural random walk this leads us to a barrier for the hardness result the non uniqueness of infinite gibbs measure is not realizable by any finite gadgets

the complexity of approximately counting in 2 spin systems on k uniform bounded degree hypergraphs one of the most important recent developments in the complexity of approximate counting is the classification of the complexity of approximating the partition functions of antiferromagnetic 2 spin systems on bounded degree graphs this classification is based on a beautiful connection to the so called uniqueness phase transition from statistical physics on the infinite regular tree our objective is to study the impact of this classification on unweighted 2 spin models on k uniform hypergraphs as has already been indicated by yin and zhao the connection between the uniqueness phase transition and the complexity of approximate counting breaks down in the hypergraph setting nevertheless we show that for every non trivial symmetric k ary boolean function f there exists a degree bound 0 so that for all 0 the following problem is np hard given a k uniform hypergraph with maximum degree at most approximate the partition function of the hypergraph 2 spin model associated with f it is np hard to approximate this partition function even within an exponential factor by contrast if f is a trivial symmetric boolean function e g any function f that is excluded from our result then the partition function of the corresponding hypergraph 2 spin model can be computed exactly in polynomial time

and or search spaces for graphical models the paper introduces an and or search space perspective for graphical models that include probabilistic networks directed or undirected and constraint networks in contrast to the traditional or search space view the and or search tree displays some of the independencies present in the graphical model explicitly and may sometimes reduce the search space exponentially indeed most algorithmic advances in search based constraint processing and probabilistic inference can be viewed as searching an and or search tree or graph familiar parameters such as the depth of a spanning tree treewidth and pathwidth are shown to play a key role in characterizing the effect of and or search graphs vs the traditional or search graphs we compare memory intensive and or graph search with inference methods and place various existing algorithms within the and or search space

on counting independent sets in sparse graphs we prove two results concerning approximate counting of independent sets in graphs with constant maximum degree delta the first implies that the markov chain monte carlo technique is likely to fail if delta geq 6 the second shows that no fully polynomial randomized approximation scheme can exist for delta geq 25 unless mathrm rp mathrm np

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

deterministic algorithms for the lovasz local lemma the lovasz local lemma lll p erdos and l lovasz problems and results on 3 chromatic hypergraphs and some related questions in infinite and finite sets vol ii a hajnal r rado and v t sos eds north holland amsterdam 1975 pp 609 627 is a powerful result in probability theory that informally states the following the probability that none of a set of bad events happens is positive if the probability of each event is small compared to the number of events that depend on it the lll is often used for nonconstructive existence proofs of combinatorial structures a prominent application is to k cnf formulas where the lll implies that if every clause in a formula shares variables with at most d leq 2 k e 1 other clauses then such a formula has a satisfying assignment recently a randomized algorithm to efficiently construct a satisfying assignment in this setting was given by moser a constructive proof of the lovasz local lemma in stoc 09 proceedings of the 41st annual acm sym

iterative state space reduction for flexible computation abstract flexible computation is a general framework for decision making under limited computational resources it enables an agent to allocate limited computational resources to maximize its overall performance or utility in this paper we present a strategy for flexible computation which we call iterative state space reduction the main ideas are to reduce a problem space that is difficult to search to one that is relatively easy to explore to use the optimal solution from the reduced space as an approximate solution to the original problem and to iteratively apply multiple reductions to progressively find better solutions the basic operation for state space reduction is heuristic node pruning which excludes the nodes that do not seem to lead to high quality solutions from further examination iterative state space reduction can be combined with a state space search such as best first search or depth first branch and bound dfbnb the resulting algorithm can run as an anytime algorithm which provides a tradeoff between solution quality and computation based on an analytical modal we analyze the probability that one iteration of the iterative process finds a solution furthermore by combining it within dfbnb we apply iterative state space reduction to three combinatorial problems the maximum boolean satisfiability the symmetric tsp and asymmetric tsp our experimental results show that iterative state space reduction is effective and efficient making dfbnb find better solutions with less computation

uniform sampling through the lovasz local lemma we propose a new algorithmic framework called partial rejection sampling to draw samples exactly from a product distribution conditioned on none of a number of bad events occurring our framework builds new connections between the variable framework of the lovasz local lemma and some classical sampling algorithms such as the cycle popping algorithm for rooted spanning trees among other applications we discover new algorithms to sample satisfying assignments of k cnf formulas with bounded variable occurrences