probability density distribution of random line segments inside a convex body application to random media we study the probability density distribution of random line segments entirely contained inside a convex body of rn relations between this distribution function and other distribution functions of importance in geometric probability are shown the expression for the probability density distribution of random line segments is given for several simple geometric bodies in r2 r3 and rn application to random media is then presented

chord length distribution function for two phase random media a statistical correlation function of basic importance in the study of two phase random media such as suspensions porous media and composites is the chord length distribution function p z we show that p z is related to another fundamentally important morphological descriptor studied by us previously namely the lineal path function l z which gives the probability of finding a line segment of length z wholly in one of the phases when randomly thrown into the sample we derive exact series representations of the chord length distribution function for media comprised of spheres with a polydispersivity in size for arbitrary space dimension d for the special case of spatially uncorrelated spheres i e fully penetrable spheres we determine exactly p z and the mean chord length mathit l mathit c the first moment of p z we also obtain corresponding formulas for the case of impenetrable i e spatially correlated polydispersed spheres

circuit satisfiability and constraint satisfaction around skolem arithmetic abstract we study interactions between skolem arithmetic and certain classes of circuit satisfiability and constraint satisfaction problems csps we revisit results of glaser et al 1 in the context of csps and settle the major open question from that paper finding a certain satisfiability problem on circuits involving complement intersection union and multiplication to be decidable this we prove using the decidability of skolem arithmetic then we solve a second question left open in 1 by proving a tight upper bound for the similar circuit satisfiability problem involving just intersection union and multiplication we continue by studying first order expansions of skolem arithmetic without constants n as csps we find already here a rich landscape of problems with non trivial instances that are in p as well as those that are np complete

distribution of distance in the spheroid the distribution of distance in the sphere is reviewed the distribution of distance in the ellipsoid is given as an integral which can be done in terms of elementary functions for the spheroid as an application maclaurin s ratio of the polar to equatorial radius of the earth due to its rotation is rederived using the distribution found here

hard constraint satisfaction problems have hard gaps at location 1 an instance of the maximum constraint satisfaction problem max csp is a finite collection of constraints on a set of variables and the goal is to assign values to the variables that maximises the number of satisfied constraints max csp captures many well known problems such as maxk sat and max cut and is consequently np hard thus it is natural to study how restrictions on the allowed constraint types or constraint language affect the complexity and approximability of max csp the pcp theorem is equivalent to the existence of a constraint language for which max csp has a hard gap at location 1 i e it is np hard to distinguish between satisfiable instances and instances where at most some constant fraction of the constraints are satisfiable all constraint languages for which the csp problem i e the problem of deciding whether all constraints can be satisfied is currently known to be np hard have a certain algebraic property we prove that any constraint language with this algebraic property makes max csp have a hard gap at location 1 which in particular implies that such problems cannot have a ptas unless p np we then apply this result to max csp restricted to a single constraint type this class of problems contains for instance max cut and max dicut assuming p np we show that such problems do not admit ptas except in some trivial cases our results hold even if the number of occurrences of each variable is bounded by a constant finally we give some applications of our results

the complexity of general valued csps an instance of the valued constraint satisfaction problem vcsp is given by a finite set of variables a finite domain of labels and a sum of functions each function depending on a subset of the variables each function can take finite values specifying costs of assignments of labels to its variables or the infinite value which indicates an infeasible assignment the goal is to find an assignment of labels to the variables that minimizes the sum we study assuming that p ne np how the complexity of this very general problem depends on the set of functions allowed in the instances the so called constraint language the case when all allowed functions take values in 0 infty corresponds to ordinary csps where one deals only with the feasibility issue and there is no optimization this case is the subject of the algebraic csp dichotomy conjecture predicting for which constraint languages csps are tractable i e solvable in polynomial time and for which they are np hard the case when all

classifying the complexity of constraints using finite algebras many natural combinatorial problems can be expressed as constraint satisfaction problems this class of problems is known to be np complete in general but certain restrictions on the form of the constraints can ensure tractability here we show that any set of relations used to specify the allowed forms of constraints can be associated with a finite universal algebra and we explore how the computational complexity of the corresponding constraint satisfaction problem is connected to the properties of this algebra hence we completely translate the problem of classifying the complexity of restricted constraint satisfaction problems into the language of universal algebra we introduce a notion of tractable algebra and investigate how the tractability of an algebra relates to the tractability of the smaller algebras which may be derived from it including its subalgebras and homomorphic images this allows us to reduce significantly the types of algebras which need to be classified using our results we also show that if the decision problem associated with a given collection of constraint types can be solved efficiently then so can the corresponding search problem we then classify all finite strictly simple surjective algebras with respect to tractability obtaining a dichotomy theorem which generalizes schaefer s dichotomy for the generalized satisfiability problem finally we suggest a possible general algebraic criterion for distinguishing the tractable and intractable cases of the constraint satisfaction problem

weak near unanimity functions and digraph homomorphism problems for a fixed digraph h the problem of deciding whether there exists a homomorphism from an input digraph g to h is known as the h colouring problem an algebraic approach to this problem was pioneered by jeavons et al in the context of the more general constraint satisfaction problem results by larose and zadori and by maroti and mckenzie allow one to interpret the algebraic approach in terms of so called weak near unanimity functions wnufs in this paper we focus on weak near unanimity functions and how they apply to the h colouring problem in particular our results range from non existence results of wnufs for certain digraphs h to the existence of wnufs for the well known polynomial cases of the h colouring problem treated by gutjahr woeginger and welzl along the way we develop wnuf analogs of the indicator and sub indicator constructions of hell and nesetril these results provide evidence to the conjecture that weak near unanimity functions are the right measure for the complexity of the h colouring problem

an algebraic theory of complexity for discrete optimization discrete optimization problems arise in many different areas and are studied under many different names in many such problems the quantity to be optimized can be expressed as a sum of functions of a restricted form here we present a unifying theory of complexity for problems of this kind we show that the complexity of a finite domain discrete optimization problem is determined by certain algebraic properties of the objective function which we call weighted polymorphisms we define a galois connection between sets of rational valued functions and sets of weighted polymorphisms and show how the closed sets of this galois connection can be characterized these results provide a new approach to studying the complexity of discrete optimization we use this approach to identify certain maximal tractable subproblems of the general problem and hence derive a complete classification of complexity for the boolean case

h coloring dichotomy revisited the h coloring problem can be expressed as a particular case of the constraint satisfaction problem csp whose computational complexity has been intensively studied under various approaches in the last several years we show that the dichotomy theorem proved by hell and neset il on the complexity of h coloring j combin theory ser b 48 1990 92 110 for the complexity of the h coloring problem for undirected graphs can be obtained using general methods for studying csp and that the criterion distinguishing the tractable cases of the h coloring problem agrees with that conjectured in a a bulatov p g jeavons a a krokhin constraint satisfaction problems and finite algebras in proc 27th internat colloq on automata languages and programming icalp 00 lecture notes in computer science vol 1853 springer berlin 2000 pp 272 282 for the complexity of the general csp

a dichotomy theorem for constraint satisfaction problems on a 3 element set the constraint satisfaction problem csp provides a common framework for many combinatorial problems the general csp is known to be np complete however certain restrictions on a possible form of constraints may affect the complexity and lead to tractable problem classes there is therefore a fundamental research direction aiming to separate those subclasses of the csp that are tractable and those which remain np complete schaefer gave an exhaustive solution of this problem for the csp on a 2 element domain in this article we generalise this result to a classification of the complexity of the csp on a 3 element domain the main result states that every subproblem of the csp is either tractable or np complete and the criterion separating them is that conjectured in bulatov et al 2005 and bulatov and jeavons 2001b we also characterize those subproblems for which standard constraint propagation techniques provide a decision procedure finally we exhibit a polynomial time algorithm which for a given set of allowed constraints outputs if this set gives rise to a tractable problem class to obtain the main result and the algorithm we extensively use the algebraic technique for the csp developed in jeavons 1998b bulatov et al 2005 and bulatov and jeavons 2001b

measurement error in the explanatory variable of a binary regression regression calibration and integrated conditional likelihood in studies of residential radon and lung cancer in epidemiology one approach to investigating the dependence of disease risk on an explanatory variable in the presence of several confounding variables is by fitting a binary regression using a conditional likelihood thus eliminating the nuisance parameters when the explanatory variable is measured with error the estimated regression coefficient is biased usually towards zero motivated by the need to correct for this bias in analyses that combine data from a number of case control studies of lung cancer risk associated with exposure to residential radon two approaches are investigated both employ the conditional distribution of the true explanatory variable given the measured one the method of regression calibration uses the expected value of the true given measured variable as the covariate the second approach integrates the conditional likelihood numerically by sampling from the distribution of the true given measured explanatory variable the two approaches give very similar point estimates and confidence intervals not only for the motivating example but also for an artificial data set with known properties these results and some further simulations that demonstrate correct coverage for the confidence intervals suggest that for studies of residential radon and lung cancer the regression calibration approach will perform very well so that nothing more sophisticated is needed to correct for measurement error

the power of linear programming for general valued csps let d called the domain be a fixed finite set and let gamma called the valued constraint language be a fixed set of functions of the form f d m to mathbb q cup infty where different functions might have different arity m we study the valued constraint satisfaction problem parametrized by gamma denoted by vcsp gamma these are minimization problems given by n variables and the objective function given by a sum of functions from gamma each depending on a subset of the n variables for example if d 0 1 and gamma contains all ternary 0 infty valued functions vcsp gamma corresponds to 3 sat more generally if gamma contains only 0 infty valued functions vcsp gamma corresponds to csp gamma if d 0 1 and gamma contains all ternary 0 1 valued functions vcsp gamma corresponds to min 3 sat in which the goal is to minimize the number of unsatisfied clauses in a 3 cnf instance finite valued constraint languages contain functions that take on only rational values and not infinite values our main result is a precise algebraic characterization of valued constraint languages whose instances can be solved exactly by the basic linear programming relaxation blp for a valued constraint language gamma blp is a decision procedure for gamma if and only if gamma admits a symmetric fractional polymorphism of every arity for a finite valued constraint language gamma blp is a decision procedure if and only if gamma admits a symmetric fractional polymorphism of some arity or equivalently if gamma admits a symmetric fractional polymorphism of arity 2 using these results we obtain tractability of several novel classes of problems including problems over valued constraint languages that are 1 submodular on arbitrary lattices 2 k submodular on arbitrary finite domains 3 weakly and hence strongly tree submodular on arbitrary trees

gap theorems for robust satisfiability boolean csps and beyond abstract a computational problem exhibits a gap property when there is no tractable boundary between two disjoint sets of instances we establish a gap trichotomy theorem for a family of constraint problem variants completely classifying the complexity of possible np hard gaps in the case of boolean domains as a consequence we obtain a number of dichotomies for the complexity of specific variants of the constraint satisfaction problem all are either polynomial time tractable or np complete schaefer s original dichotomy for sat variants is a notable particular case universal algebraic methods have been central to recent efforts in classifying the complexity of constraint satisfaction problems a second contribution of the article is to develop aspects of the algebraic approach in the context of a number of variants of the constraint satisfaction problem in particular this allows us to lift our results on boolean domains to many templates on non boolean domains

duality and polynomial testing of tree homomorphisms let h be a fixed digraph we consider the h colouring problem i e the problem of deciding which digraphs g admit a homomorphism to h we are interested in a characterization in terms of the absence in g of certain tree like obstructions specifically we say that h has tree duality if for all digraphs g g is not homomorphic to h if and only if there is an oriented tree which is homomorphic to g but not to h we prove that if h has tree duality then the h colouring problem is polynomial we also generalize tree duality to bounded treewidth duality and prove a similar result we relate these duality concepts to the notion of the x property studied by gutjahr welzl and woeginger we then focus on the case when h itself is an oriented tree in fact we are particularly interested in those trees that have exactly one vertex of degree three and all other vertices of degree one or two such trees are called triads we have shown in a companion paper that there exist oriented triads h for which the h colouring problem is np complete we contrast these with several families of oriented triads h which have tree duality or bounded treewidth duality and hence polynomial h colouring problems if p 6 np then no oriented triad h with an np complete h colouring problem can have bounded treewidth duality however no proof of this is known for any oriented triad h we prove that none of the oriented triads h with np complete hcolouring problems given in the companion paper has tree duality

the complexity of counting quantifiers on equality languages an equality language is a relational structure with infinite domain whose relations are first order definable in equality we classify the extensions of the quantified constraint satisfaction problem over equality languages in which the native existential and universal quantifiers are augmented by some subset of counting quantifiers in doing this we find ourselves in various worlds in which dichotomies or trichotomies subsist

the performance of functional methods for correcting non gaussian measurement error within poisson regression corrected excess risk of lung cancer mortality in relation to radon exposure among french uranium miners a broad variety of methods for measurement error me correction have been developed but these methods have rarely been applied possibly because their ability to correct me is poorly understood we carried out a simulation study to assess the performance of three error correction methods two variants of regression calibration the substitution method and the estimation calibration method and the simulation extrapolation simex method features of the simulated cohorts were borrowed from the french uranium miners cohort in which exposure to radon had been documented from 1946 to 1999 in the absence of me correction we observed a severe attenuation of the true effect of radon exposure with a negative relative bias of the order of 60 on the excess relative risk of lung cancer death in the main scenario considered that is when me characteristics previously determined as most plausible from the french uranium miners cohort were used both to generate exposure data and to correct for me at the analysis stage all three error correction methods showed a noticeable but partial reduction of the attenuation bias with a slight advantage for the simex method however the performance of the three correction methods highly depended on the accurate determination of the characteristics of me in particular we encountered severe overestimation in some scenarios with the simex method and we observed lack of correction with the three methods in some other scenarios for illustration we also applied and compared the proposed methods on the real data set from the french uranium miners cohort study copyright 2012 john wiley sons ltd

covariate measurement errors and parameter estimation in a failure time regression model summary estimation in cox s failure time regression model is considered when the regression vector is subject to measurement error a hazard function model is induced for the failure rate given the measured covariate and a partial likelihood function is derived for the relative risk parameters this partial likelihood function may involve the baseline hazard function as well as the regression parameter but useful inference techniques arise for testing whether the regression parameter equals zero and for more general inferences in important special cases explicit consideration is given to testing equality of survival curves when group membership is subject to misclassification and to relative risk estimation with normally distributed covariates approaches to regression estimation using the overall likelihood function and a marginal likelihood function based on failure time ranks are also indicated illustration of the possible effect of covariate errors on relative risk estimation is provided

congruence distributivity implies bounded width we show that a constraint language with compatible jonsson terms or equivalently associated with an algebra generating a congruence distributive variety defines a constraint satisfaction problem solvable by the local consistency checking algorithm

correction of logistic regression relative risk estimates and confidence intervals for systematic within person measurement error errors in the measurement of exposure that are independent of disease status tend to bias relative risk estimates and other measures of effect in epidemiologic studies toward the null value two methods are provided to correct relative risk estimates obtained from logistic regression models for measurement errors in continuous exposures within cohort studies that may be due to either random unbiased within person variation or to systematic errors for individual subjects these methods require a separate validation study to estimate the regression coefficient lambda relating the surrogate measure to true exposure in the linear approximation method the true logistic regression coefficient beta is estimated by beta lambda where beta is the observed logistic regression coefficient based on the surrogate measure in the likelihood approximation method a second order taylor series expansion is used to approximate the logistic function enabling closed form likelihood estimation of beta confidence intervals for the corrected relative risks are provided that include a component representing error in the estimation of lambda based on simulation studies both methods perform well for true odds ratios up to 3 0 for higher odds ratios the likelihood approximation method was superior with respect to both bias and coverage probability an example is provided based on data from a prospective study of dietary fat intake and risk of breast cancer and a validation study of the questionnaire used to assess dietary fat intake

application of geometric probability techniques to the evaluation of interaction energies arising from a general radial potential a formalism is developed for using geometric probability techniques to evaluate interaction energies arising from a general radial potential v r12 where r12 r2 r1 the integrals that arise in calculating these energies can be separated into a radial piece that depends on r12 and a nonradial piece that describes the geometry of the system including the density distribution we show that all geometric information can be encoded into a radial density function g r12 1 2 which depends on r12 and the densities 1 and 2 of two interacting regions g r12 1 2 is calculated explicitly for several geometries and is then used to evaluate interaction energies for several cases of interest our results find application in elementary particle nuclear and atomic physics

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

some aspects of measurement error in explanatory variables for continuous and binary regression models a simple form of measurement error model for explanatory variables is studied incorporating classical and berkson cases as particular forms and allowing for either additive or multiplicative errors the work is motivated by epidemiological problems and therefore consideration is given not only to continuous response variables but also to logistic regression models the possibility that different individuals in a study have errors of different types is also considered the relatively simple estimation procedures proposed for use with cohort data and case control data are checked by simulation under the assumption of various error structures the results show that even in situations where conventional analysis yields slope estimates that are on average attenuated by a factor of approximately 50 per cent estimates obtained using the proposed amended likelihood functions are within 5 per cent of their true values the work was carried out to provide a method for the analysis of lung cancer risk following residential radon exposure but it should be applicable to a wide variety of situations

the complexity of conservative valued csps we study the complexity of valued constraint satisfaction problems vcsps parametrized by a constraint language a fixed set of cost functions over a finite domain an instance of the problem is specified by a sum of cost functions from the language and the goal is to minimize the sum under the unique games conjecture the approximability of finite valued vcsps is well understood see raghavendra 2008 however there is no characterization of finite valued vcsps let alone general valued vcsps that can be solved exactly in polynomial time thus giving insights from a combinatorial optimization perspective we consider the case of languages containing all possible unary cost functions in the case of languages consisting of only 0 valued cost functions i e relations such languages have been called conservative and studied by bulatov 2003 2011 and recently by barto 2011 since we study valued languages we call a language conservative if it contains all finite valued unary cost functions the computational complexity of conservative valued languages has been studied by cohen et al 2006 for languages over boolean domains by deineko et al 2008 for 0 1 valued languages a k a max csp and by takhanov 2010a for 0 valued languages containing all finite valued unary cost functions a k a min cost hom we prove a schaefer like dichotomy theorem for conservative valued languages if all cost functions in the language satisfy a certain condition specified by a complementary combination of stp and mjn multimorphisms then any instance can be solved in polynomial time via a new algorithm developed in this article otherwise the language is np hard this is the first complete complexity classification of general valued constraint languages over non boolean domains it is a common phenomenon that complexity classifications of problems over non boolean domains are significantly harder than the boolean cases the polynomial time algorithm we present for the tractable cases is a generalization of the submodular minimization problem and a result of cohen et al 2008 our results generalize previous results by takhanov 2010a and a subset of results by cohen et al 2006 and deineko et al 2008 moreover our results do not rely on any computer assisted search as in deineko et al 2008 and provide a powerful tool for proving hardness of finite valued and general valued languages

closure properties of constraints many combinatorial search problems can be expressed as constraint satisfaction problems and this class of problems is known to be np complete in general in this paper we investigate the subclasses that arise from restricting the possible constraint types we first show that any set of constraints that does not give rise to an np complete class of problems must satisfy a certain type of algebraic closure condition we then investigate all the different possible forms of this algebraic closure property and establish which of these are sufficient to ensure tractability as examples we show that all known classes of tractable constraints over finite domains can be characterized by such an algebraic closure property finally we describe a simple computational procedure that can be used to determine the closure properties of a given set of constraints this procedure involves solving a particular constraint satisfaction problem which we call an indicator problem

the complexity of finite valued csps we study the computational complexity of exact minimization of rational valued discrete functions let be a set of rational valued functions on a fixed finite domain such a set is called a finite valued constraint language the valued constraint satisfaction problem vcsp is the problem of minimizing a function given as a sum of functions from we establish a dichotomy theorem with respect to exact solvability for all finite valued constraint languages defined on domains of arbitrary finite size we show that every constraint language either admits a binary symmetric fractional polymorphism in which case the basic linear programming relaxation solves any instance of vcsp exactly or satisfies a simple hardness condition that allows for a polynomial time reduction from max cut to vcsp

a proof of the csp dichotomy conjecture many natural combinatorial problems can be expressed as constraint satisfaction problems this class of problems is known to be np complete in general but certain restrictions on the form of the constraints can ensure tractability the standard way to parameterize interesting subclasses of the constraint satisfaction problem is via finite constraint languages the main problem is to classify those subclasses that are solvable in polynomial time and those that are np complete it was conjectured that if a constraint language has a weak near unanimity polymorphism then the corresponding constraint satisfaction problem is tractable otherwise it is np complete in the article we present an algorithm that solves constraint satisfaction problem in polynomial time for constraint languages having a weak near unanimity polymorphism which proves the remaining part of the conjecture 1

simulation extrapolation estimation in parametric measurement error models we describe a simulation based method of inference for parametric measurement error models in which the measurement error variance is known or at least well estimated the method entails adding add

the computational structure of monotone monadic snp and constraint satisfaction a study through datalog and group theory this paper starts with the project of finding a large subclass of np which exhibits a dichotomy the approach is to find this subclass via syntactic prescriptions while the paper does not achieve this goal it does isolate a class of problems specified by monotone monadic snp without inequality which may exhibit this dichotomy we justify the placing of all these restrictions by showing essentially using ladner s theorem that classes obtained by using only two of the above three restrictions do not show this dichotomy we then explore the structure of this class we show that all problems in this class reduce to the seemingly simpler class csp we divide csp into subclasses and try to unify the collection of all known polytime algorithms for csp problems and extract properties that make csp problems np hard this is where the second part of the title a study through datalog and group theory comes in we present conjectures about this class which would end in showing the dichotomy

the csp dichotomy holds for digraphs with no sources and no sinks a positive answer to a conjecture of bang jensen and hell bang jensen and hell conjectured in 1990 using the language of graph homomorphisms a constraint satisfaction problem csp dichotomy for digraphs with no sources or sinks the conjecture states that the csp for such a digraph is tractable if each component of its core is a cycle and is np complete otherwise in this paper we prove this conjecture and as a consequence a conjecture of bang jensen hell and macgillivray from 1995 classifying hereditarily hard digraphs further we show that the csp dichotomy for digraphs with no sources or sinks agrees with the algebraic characterization conjectured by bulatov jeavons and krokhin in 2005

constraint satisfaction problems over semilattice block mal tsev algebras abstract the dichotomy conjecture for the constraint satisfaction problem csp was recently settled independently by zhuk and the author the proofs of this conjecture are rather sophisticated and require deep understanding of the algebraic structure of csps this paper is a precursor of the author s proof of the dichotomy conjecture and represents its main ideas in a simpler and clearer form in a more restricted class of the csp there are two well known types of algorithms for solving csps local propagation and generating a basis of the solution space for several years the focus of the csp research has been on hybrid algorithms that somehow combine the two approaches in this paper we present a new method of such hybridization that allows us to solve certain csps that has been out of reach for a quite a while and eventually leads to resolving the dichotomy conjecture we apply this method to csps parametrized by a universal algebra an approach that has been very popular in the last decade or so specifically we consider a fairly restricted class of algebras we will call semilattice block mal tsev an algebra a is semilattice block mal tsev if it has a binary operation f a ternary operation m and a congruence such that the quotient a with operation f is a semilattice f is a projection on every block of and every block of is a mal tsev algebra with mal tsev operation m this means that the domain in such a csp is partitioned into blocks such that if the problem is considered on the quotient set a it can be solved by a simple constraint propagation algorithm on the other hand if the problem is restricted on individual blocks it can be solved by generating a basis of the solution space we show that the two methods can be combined in a highly nontrivial way and therefore the constraint satisfaction problem over a semilattice block mal tsev algebra is solvable in polynomial time

covariate measurement error in logistic regression abstract in a logistic regression model when covariates are subject to measurement error the naive estimator obtained by regressing on the observed covariates is asymptotically biased this document introduces a bias adjusted estimator and two estimators appropriate for normally distributed measurement errors a functional maximum likelihood estimator and an estimator which exploits the consequences of sufficiency the four proposals are studied asymptotically under conditions which are appropriate when the measurement error is small a small monte carlo study illustrates the superiority of the measurement error estimators in certain situations additional keywords mathematical models

complexity of counting csp with complex weights we give a complexity dichotomy theorem for the counting constraint satisfaction problem csp in short with algebraic complex weights to this end we give three conditions for its tractability let f be any finite set of algebraic complex valued functions defined on an arbitrary finite domain we show that csp f is solvable in polynomial time if all three conditions are satisfied and is p hard otherwise our dichotomy theorem generalizes a long series of important results on counting problems and reaches a natural culmination a the problem of counting graph homomorphisms is the special case when f has a single symmetric binary function dyer and greenhill 2000 bulatov and grohe 2005 goldberg et al 2010 cai et al 2013 b the problem of counting directed graph homomorphisms is the special case when f has a single but not necessarily symmetric binary function dyer et al 2007 cai and chen 2010 c the unweighted form of csp is when all functions in f take values in 0 1 bulatov 2008 dyer and richerby 2013