computational complexity of holant problems we propose and explore a novel alternative framework to study the complexity of counting problems called holant problems compared to counting constraint satisfaction problems csp it is a refinement with a more explicit role for the constraint functions both graph homomorphism and csp can be viewed as special cases of holant problems we prove complexity dichotomy theorems in this framework our dichotomy theorems apply to local constraint functions which are symmetric functions on boolean input variables and evaluate to arbitrary real or complex values we discover surprising tractable subclasses of counting problems which could not easily be specified in the csp framework when all unary functions are assumed to be free mathrm holant problems the tractable ones consist of functions that are degenerate or of arity at most two or holographic transformations of fibonacci gates when only two special unary functions the constant zero and constant one functions are assumed to be free mathrm holant c problems we further identify three special families of tractable cases then we prove that all other cases are p hard the main technical tool we use and develop is holographic reductions another technical tool used in combination with holographic reductions is polynomial interpolations

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

an effective dichotomy for the counting constraint satisfaction problem bulatov proceedings of the 35 th international colloquium on automata languages and programming part 1 lecture notes in comput sci 5125 springer new york 2008 pp 646 661 gave a dichotomy for the counting constraint satisfaction problem csp a problem from csp is characterized by a constraint language gamma a fixed finite set of relations over a finite domain d an instance of the problem uses these relations to constrain an arbitrarily large finite set of variables bulatov showed that the problem of counting the satisfying assignments of instances of any problem from csp is either in polynomial time fp or is p complete his proof draws heavily on techniques from universal algebra and cannot be understood without a secure grasp of that field we give an elementary proof of bulatov s dichotomy based on succinct representations which we call frames of a class of highly structured relations which we call strongly rectangular we show that these are precisely the relation

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

generalizing constraint satisfaction on trees hybrid tractability and variable elimination the constraint satisfaction problem csp is a central generic problem in artificial intelligence considerable progress has been made in identifying properties which ensure tractability in such problems such as the property of being tree structured in this paper we introduce the broken triangle property which allows us to define a novel tractable class for this problem which significantly generalizes the class of problems with tree structure we show that the broken triangle property is conservative i e it is preserved under domain reduction and hence under arc consistency operations and that there is a polynomial time algorithm to determine an ordering of the variables for which the broken triangle property holds or to determine that no such ordering exists we also present a non conservative extension of the broken triangle property which is also sufficient to ensure tractability and can also be detected in polynomial time we show that both the broken triangle property and its extension can be used to eliminate variables and that both of these properties provide the basis for preprocessing procedures that yield unique closures orthogonal to value elimination by enforcement of consistency finally we also discuss the possibility of using the broken triangle property in variable ordering heuristics

on the cohen macaulay type of s lines in an 1 abstract let a be the coordinate ring of s lines through the origin in a n 1 k we discuss what it means for these lines to be in sufficiently general position and then with this restriction on the lines we attempt to verify our belief that the cohen macaulay type of a depends only on s and n we characterize those s and n for which a is a gorenstein ring i e of c m type 1 and explicitly calculate the cohen macaulay type in several other cases we then extend some of our results to the case of an arbitrary reduced curve whose tangent cone consists of lines in sufficiently general position finally we calculate the hubert samuel polynomials of the curves we have been considering

brownian motion in cones we study the asymptotic behavior of brownian motion and its conditioned process in cones using an infinite series representation of its transition density a concise probabilistic interpretation of this series in terms of the skew product decomposition of brownian motion is derived and used to show properties of the transition density

combinatorial problems raised from 2 semilattices abstract the constraint satisfaction problem csp provides a general framework for many combinatorial problems in a a bulatov a a krokhin p g jeavons classifying the complexity of constraints using finite algebras siam j comput 34 3 2005 720 742 p g jeavons on the algebraic structure of combinatorial problems theoret comput sci 200 1998 185 204 and then in a a bulatov p g jeavons algebraic structures in combinatorial problems technical report math al 4 2001 technische universitat dresden dresden germany 2001 a new approach to the study of the csp has been developed which uses properties of universal algebras assigned to certain subclasses of the csp such that the time complexity and other properties of subclasses can be derived from the properties of the assigned algebras in this paper we briefly survey this approach and then prove that problem classes corresponding to finite 2 semilattices that is groupoids satisfying the identities x x x x y y x x x y x x y can be solved in polynomial time making use of this result we classify finite conservative groupoids and 4 element algebras with minimal clone of term operations with respect to the complexity of the corresponding csp class

heat kernel expansion on a generalized cone it is shown that the short time expansion of the integrated heat kernel on a locally flat generalized cone c n as defined by cheeger consists of just the weyl volume term and the constant term this latter is explicitly evaluated when n is a lens space sd zm for odd d elliptic space sd z2 for all d and any of the three dimensional homogeneous space forms s3 agreement is found with the corresponding expansion on the orbifold version t4 z2 of the k3 surface and in fact with all td z2

generalising submodularity and horn clauses tractable optimization problems defined by tournament pair multimorphisms the submodular function minimization problem sfm is a fundamental problem in combinatorial optimization and several fully combinatorial polynomial time algorithms have recently been discovered to solve this problem the most general versions of these algorithms are able to minimize any submodular function whose domain is a set of tuples over any totally ordered finite set and whose range includes both finite and infinite values in this paper we demonstrate that this general form of sfm is just one example of a much larger class of tractable discrete optimization problems defined by valued constraints these tractable problems are characterized by the fact that their valued constraints have an algebraic property which we call a tournament pair multimorphism this larger tractable class also includes the problem of satisfying a set of horn clauses horn sat as well as various extensions of this problem to larger finite domains

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

a generic arc consistency algorithm and its specializations consistency techniques have been studied extensively in the past as a way of tackling constraint satisfaction problems csp in particular various arc consistency algorithms have been proposed originating from waltz s filtering algorithm and culminating in the optimal algorithm ac 4 of mohr and henderson ac 4 runs in o ed 2 in the worst case where e is the number of arcs or constraints and d is the size of the largest domain being applicable to the whole class of binary csp these algorithms do not take into account the semantics of constraints in this paper we present a new generic arc consistency algorithm ac 5 the algorithm is parameterized on two specified procedures and can be instantiated to reduce to ac 3 and ac 4 more important ac 5 can be instantiated to produce an o ed 2 algorithm for a number of important classes of constraints functional anti functional monotonic and their generalization to functional anti functional and monotonic piecewise constraints we also show that ac 5 has an important application in constraint logic programming over finite domains the kernel of the constraint solver for such a programming language is an arc consistency algorithm for a set of basic constraints we prove that ac 5 in conjuction with node consistency provides a decision procedure for these constraints running in time o ed

constraint satisfaction problems solvable by local consistency methods we prove that constraint satisfaction problems without the ability to count are solvable by the local consistency checking algorithm this settles three equivalent conjectures feder vardi sicomp 98 bulatov lics 04 and larose zadori au 07

a comparison of structural csp decomposition methods abstract we compare tractable classes of constraint satisfaction problems csps we first give a uniform presentation of the major structural csp decomposition methods we then introduce a new class of tractable csps based on the concept of hypertree decomposition recently developed in database theory and analyze the cost of solving csps having bounded hypertree width we provide a framework for comparing parametric decomposition based methods according to tractability criteria and compare the most relevant methods we show that the method of hypertree decomposition dominates the others in the case of general csps i e csps of unbounded arity we also make comparisons for the restricted case of binary csps finally we consider the application of decomposition methods to the dual graph of a hypergraph in fact this technique is often used to exploit binary decomposition methods for nonbinary csps however even in this case the hypertree decomposition method turns out to be the most general method

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

the complexity of quantified constraint satisfaction collapsibility sink algebras and the three element case the constraint satisfaction probem csp is a well acknowledged framework in which many combinatorial search problems can be naturally formulated the csp may be viewed as the problem of deciding the truth of a logical sentence consisting of a conjunction of constraints in front of which all variables are existentially quantified the quantified constraint satisfaction problem qcsp is the generalization of the csp where universal quantification is permitted in addition to existential quantification the general intractability of these problems has motivated research studying the complexity of these problems under a restricted constraint language which is a set of relations that can be used to express constraints this paper introduces collapsibility a technique for deriving positive complexity results on the qcsp in particular this technique allows one to show that for a particular constraint language the qcsp reduces to the csp we show that collapsibility applies to three known tractable cases of the qcsp that were originally studied using disparate proof techniques in different decades quantified 2 sat aspvall plass and tarjan in 1979 quantified horn sat karpinski kleine buning and schmitt in 1987 and quantified affine sat creignou khanna and sudan in 2001 this reconciles and reveals common structure among these cases which are describable by constraint languages over a two element domain in addition to unifying these known tractable cases we study constraint languages over domains of larger size

spectral functions of non essentially self adjoint operators fil falomir horacio alberto consejo nacional de investigaciones cientificas y tecnicas centro cientifico tecnologico conicet la plata instituto de fisica la plata universidad nacional de la plata facultad de ciencias exactas instituto de fisica la plata argentina

closed form summation of the dowker and related sums finite sums of powers of cosecants appear in a wide range of physical problems we through a unified approach which uses contour integrals and residues establish the summation formulas for two general families of such sums one of them is the family which was first studied and summed in closed form by dowker phys rev d 36 3095 1987 while the other is related to it and has not been studied before our summation formulas of the dowker sums involve only the stirling numbers of the first kind and the ordinary bernoulli polynomials and numbers unlike the earlier summation formulas in which either the higher order bernoulli numbers and polynomials or the multiple sums involving the bernoulli numbers and their products were used a great deal of other known or presumably new closed form summations follows as straightforward corollaries to these formulas among them are two special cases of the celebrated verlinde s formula and numerous sums encountered in various physical problems by mccoy and or

explicit expressions for finite trigonometric sums abstract in this paper we perform a further investigation for finite trigonometric sums we establish some connections between the higher order trigonometric functions and zeta functions as applications various known finite trigonometric sums are explicitly expressed as linear combinations of the bernoulli and euler polynomials and numbers

experimental evaluation of preprocessing algorithms for constraint satisfaction problems abstract this paper presents an experimental evaluation of two orthogonal schemes for pre processing constraint satisfaction problems csps the first of these schemes involves a class of local consistency techniques that includes directional arc consistency directional path consistency and adaptive consistency the other scheme concerns the prearrangement of variables in a linear order to facilitate an efficient search in the first series of experiments we evaluated the effect of each of the local consistency techniques on backtracking and backjumping surprisingly although adaptive consistency has the best worst case complexity bounds we have found that it exhibits the worst performance unless the constraint graph was very sparse directional arc consistency followed by either backjumping or backtracking and backjumping without any preprocessing outperformed all other techniques moreover the former dominated the latter in computationally intensive situations the second series of experiments suggests that maximum cardinality and minimum width are the best preordering i e static ordering strategies while dynamic search rearrangement is superior to all the preorderings studied

spin systems on k regular graphs with complex edge functions let k 1 be an integer and let h h 0 0 h 0 1 h 1 0 h 1 1 be a complex valued symmetric function on domain 0 1 i e where h 0 1 h 1 0 we introduce a new technique called a syzygy and prove a dichotomy theorem for the following class of problems specified by k and h given an arbitrary k regular graph g v e where the function h is attached to each edge compute z g email protected s v 0 1 u v eh s u s v z is known as the partition function of the spin system also known as counting graph homomorphisms on domain size two and is a special case of holant problems the dichotomy theorem gives a complete classification of the computational complexity of this problem depending on k and h the dependence on k and h is explicit

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

the complexity of the counting constraint satisfaction problem the counting constraint satisfaction problem csp h over a finite relational structure h can be expressed as follows given a relational structure g over the same vocabulary determine the number of homomorphisms from g to h in this article we characterize relational structures h for which csp h can be solved in polynomial time and prove that for all other structures the problem is p complete

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

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

generic expression hardness results for primitive positive formula comparison we study the expression complexity of two basic problems involving the comparison of primitive positive formulas equivalence and containment in particular we study the complexity of these problems relative to finite relational structures we present two generic hardness results for the studied problems and discuss evidence that they are optimal and yield for each of the problems a complexity trichotomy

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

an optimal k consistency algorithm abstract this paper generalizes the arc consistency algorithm of mohr and henderson 4 and the path consistency algorithm of han and lee 2 to a k consistency algorithm arc consistency and path consistency being 2 consistency and 3 consistency respectively the algorithm is a development of freuder s synthesis algorithm 1 it simultaneously establishes i consistency for each 1 i k it has worst case time and space complexity which is optimal when k is a constant and almost optimal for all other values of k in the case that all order i constraints exist for all 1 i n this algorithm is a solution to the consistent labeling problem with almost optimal worst case time and space complexity

a complexity dichotomy for partition functions with mixed signs partition functions also known as homomorphism functions form a rich family of graph invariants that contain combinatorial invariants such as the number of k colorings or the number of independent sets of a graph and also the partition functions of certain spin glass models of statistical physics such as the ising model building on earlier work by dyer and greenhill random structures algorithms 17 2000 pp 260 289 and bulatov and grohe theoret comput sci 348 2005 pp 148 186 we completely classify the computational complexity of partition functions our main result is a dichotomy theorem stating that every partition function is either computable in polynomial time or p complete partition functions are described by symmetric matrices with real entries and we prove that it is decidable in polynomial time in terms of the matrix whether a given partition function is in polynomial time or p complete while in general it is very complicated to give an explicit algebraic or combinatorial description of the tractable cases for partition functions described by hadamard matrices these turn out to be central in our proofs we obtain a simple algebraic tractability criterion which says that the tractable cases are those representable by a quadratic polynomial over the field mathbb f 2

the complexity of constraint satisfaction games and qcsp we study the complexity of two person constraint satisfaction games an instance of such a game is given by a collection of constraints on overlapping sets of variables and the two players alternately make moves assigning values from a finite domain to the variables in a specified order the first player tries to satisfy all constraints while the other tries to break at least one constraint the goal is to decide whether the first player has a winning strategy we show that such games can be conveniently represented by a logical form of quantified constraint satisfaction where an instance is given by a first order sentence in which quantifiers alternate and the quantifier free part is a conjunction of positive atomic formulas the goal is to decide whether the sentence is true while the problem of deciding such a game is pspace complete in general by restricting the set of allowed constraint predicates one can obtain infinite classes of constraint satisfaction games of lower complexity we use the quantified constraint satisfaction framework to study how the complexity of deciding such a game depends on the parameter set of allowed predicates with every predicate one can associate certain predicate preserving operations called polymorphisms we show that the complexity of our games is determined by the surjective polymorphisms of the constraint predicates we illustrate how this result can be used by identifying the complexity of a wide variety of constraint satisfaction games

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

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

the complexity of soft constraint satisfaction over the past few years there has been considerable progress in methods to systematically analyse the complexity of constraint satisfaction problems with specified constraint types one very powerful theoretical development in this area links the complexity of a set of constraints to a corresponding set of algebraic operations known as polymorphisms in this paper we extend the analysis of complexity to the more general framework of combinatorial optimisation problems expressed using various forms of soft constraints we launch a systematic investigation of the complexity of these problems by extending the notion of a polymorphism to a more general algebraic operation which we call a multimorphism we show that many tractable sets of soft constraints both established and novel can be characterised by the presence of particular multimorphisms we also show that a simple set of np hard constraints has very restricted multimorphisms finally we use the notion of multimorphism to give a complete classification of complexity for the boolean case which extends several earlier classification results for particular special cases

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

heat kernels and functional determinants on the generalized cone we consider zeta functions and heat kernel expansions on the bounded generalized cone in arbitrary dimensions using an improved calculational technique the specific case of a global monopole is analysed in detail and some restrictions thereby placed on thea 5 2 coefficient the computation of functional determinants is also addressed general formulas are given and known results are incidentally and rapidly reproduced

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

the complexity of temporal constraint satisfaction problems a temporal constraint language is a set of relations that has a first order definition in q an extended abstract of this article appeared in the proceedings of stoc 08

constructing np intermediate problems by blowing holes with parameters of various properties the search for natural np intermediate problems is one of the holy grails within computational complexity ladner s original diagonalization technique for generating np intermediate problems blowing holes has a serious shortcoming it creates problems with a highly artificial structure by arbitrarily removing certain problem instances in this article we limit this problem by generalizing ladner s method to use parameters with various characteristics this allows one to define more fine grained parameters resulting in np intermediate problems where we only blow holes in a controlled subset of the problem we begin by fully characterizing the problems that admit np intermediate subproblems for a broad and natural class of parameterizations and extend the result further such that structural csp restrictions based on parameters that are hard to compute such as tree width are covered thereby generalizing a result by grohe for studying certain classes of problems including csps parameterized by constraint languages we consider more powerful parameterizations first we identify a new method for obtaining constraint languages such that csp are np intermediate the sets can have very different properties compared to previous constructions by for instance bodirsky grohe and provides insights into the algebraic approach for studying the complexity of infinite domain csps second we prove that the propositional abduction problem parameterized by constraint languages admits np intermediate problems this settles an open question posed by nordh zanuttini we provide a framework to obtain np intermediate problems based on diagonalization we categorize different types of parameters that are usable in our framework we construct np intermediate constraint satisfaction problems we resolve the open question whether boolean abduction admits intermediate problems

on the algebraic structure of combinatorial problems abstract we describe a general algebraic formulation for a wide range of combinatorial problems including satisfiability graph colorability and graph isomorphism in this formulation each problem instance is represented by a pair of relational structures and the solutions to a given instance are homomorphisms between these relational structures the corresponding decision problem consists of deciding whether or not any such homomorphisms exist we then demonstrate that the complexity of solving this decision problem is determined in many cases by simple algebraic properties of the relational structures involved this result is used to identify tractable subproblems of satisfiability and to provide a simple test to establish whether a given set of boolean relations gives rise to one of these tractable subproblems

constraints consistency and closure although the constraint satisfaction problem is np complete in general a number of constraint classes have been identified for which some fixed level of local consistency is sufficient to ensure global consistency in this paper we describe a simple algebraic property which characterises all possible constraint types for which strong k consistency is sufficient to ensure global consistency for each k 2 we give a number of examples to illustrate the application of this result

a complete dichotomy rises from the capture of vanishing signatures we prove a complexity dichotomy theorem for holant problems over an arbitrary set of complex valued symmetric constraint functions mathcal f on boolean variables this extends and unifies all previous dichotomies for holant problems on symmetric constraint functions taking values without a finite modulus we define and characterize all symmetric vanishing signatures they turn out to be essential to the complete classification of holant problems the dichotomy theorem has an explicit tractability criterion expressible in terms of holographic transformations a holant problem defined by a set of constraint functions mathcal f is solvable in polynomial time if it satisfies this tractability criterion and is p hard otherwise the tractability criterion can be intuitively stated as follows a set mathcal f is tractable if 1 every function in mathcal f has arity at most two or 2 mathcal f is transformable to an affine type or 3 mathcal f is transformable to a product type o

tractable hypergraph properties for constraint satisfaction and conjunctive queries an important question in the study of constraint satisfaction problems csp is understanding how the graph or hypergraph describing the incidence structure of the constraints influences the complexity of the problem for binary csp instances that is where each constraint involves only two variables the situation is well understood the complexity of the problem essentially depends on the treewidth of the graph of the constraints grohe 2007 marx 2010b however this is not the correct answer if constraints with unbounded number of variables are allowed and in particular for csp instances arising from query evaluation problems in database theory formally if h is a class of hypergraphs then let csp h be csp restricted to instances whose hypergraph is in h our goal is to characterize those classes of hypergraphs for which csp h is polynomial time solvable or fixed parameter tractable parameterized by the number of variables note that in the applications related to database query evaluation we usually assume that the number of variables is much smaller than the size of the instance thus parameterization by the number of variables is a meaningful question the most general known property of h that makes csp h polynomial time solvable is bounded fractional hypertree width here we introduce a new hypergraph measure called submodular width and show that bounded submodular width of h which is a strictly more general property than bounded fractional hypertree width implies that csp h is fixed parameter tractable in a matching hardness result we show that if h has unbounded submodular width then csp h is not fixed parameter tractable and hence not polynomial time solvable unless the exponential time hypothesis eth fails the algorithmic result uses tree decompositions in a novel way instead of using a single decomposition depending on the hypergraph the instance is split into a set of instances all on the same set of variables as the original instance and then the new instances are solved by choosing a different tree decomposition for each of them the reason why this strategy works is that the splitting can be done in such a way that the new instances are uniform with respect to the number extensions of partial solutions and therefore the number of partial solutions can be described by a submodular function for the hardness result we prove via a series of combinatorial results that if a hypergraph h has large submodular width then a 3sat instance can be efficiently simulated by a csp instance whose hypergraph is h to prove these combinatorial results we need to develop a theory of multicommodity flows on hypergraphs and vertex separators in the case when the function b s defining the cost of separator s is submodular which can be of independent interest

max ones generalized to larger domains we study a family of problems called maximum solution where the objective is to maximize a linear goal function over the feasible integer assignments to a set of variables subject to a set of constraints when the domain is boolean i e restricted to 0 1 the maximum solution problem is identical to the well studied max ones problem and the approximability is completely understood for all restrictions on the underlying constraints s khanna m sudan l trevisan and d p williamson siam j comput 30 2001 pp 1863 1920 we continue this line of research by considering domains containing more than two elements we present two main results a complete classification for the approximability of all maximal constraint languages over domains of cardinality at most 4 and a complete classification of the approximability of the problem when the set of allowed constraints contains all permutation constraints under the assumption that a conjecture due to szczepara minimal clones generated by groupoids ph d thesis universite de montreal montreal qc 1996 holds we give a complete classification for all maximal constraint languages these classes of languages are well studied in universal algebra and computer science they have for instance been considered in connection with machine learning and constraint satisfaction our results are proved by using algebraic results from clone theory and the results indicate that this approach is very powerful for classifying the approximability of certain optimization problems

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

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

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

computational complexity of covering three vertex multigraphs a covering projection from a graph g onto a graph h is a mapping of the vertices of g onto the vertices of h such that for every vertex v of g the neighborhood of v is mapped bijectively onto the neighborhood of its image moreover if g and h are multigraphs then this local bijection has to preserve multiplicities of the neighbors as well the notion of covering projection stems from topology but has found applications in areas such as the theory of local computation and construction of highly symmetric graphs it provides a restrictive variant of the constraint satisfaction problem with additional symmetry constraints on the behavior of the homomorphisms of the structures involved we investigate the computational complexity of the problem of deciding the existence of a covering projection from an input graph g to a fixed target graph h among other partial results this problem has been shown np hard for simple regular graphs h of valency greater than 2 and a full characterization of computational complexity has been shown for target multigraphs with 2 vertices we extend the previously known results to the ternary case i e we give a full characterization of the computational complexity in the case of multigraphs with 3 vertices we show that even in this case a p np completeness dichotomy holds

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

closed form summations of dowker s and related trigonometric sums through a unified and relatively simple approach which uses complex contour integrals particularly convenient integration contours and calculus of residues closed form summation formulas for 12 very general families of trigonometric sums are deduced one of them is a family of cosecant sums which was first summed in closed form in a series of papers by dowker 1987 phys rev d 36 3095 101 1989 j math phys 30 770 3 1992 j phys a math gen 25 2641 8 whose method has inspired our work in this area all of the formulas derived here involve the higher order bernoulli polynomials this article is part of a special issue of journal of physics a mathematical and theoretical in honour of stuart dowker s 75th birthday devoted to applications of zeta functions and other spectral functions in mathematics and physics

the approximability of max csp with fixed value constraints in the maximum constraint satisfaction problem max csp one is given a finite collection of possibly weighted constraints on overlapping sets of variables and the goal is to assign values from a given finite domain to the variables so as to maximize the number or the total weight for the weighted case of satisfied constraints this problem is np hard in general and therefore it is natural to study how restricting the allowed types of constraints affects the approximability of the problem in this article we show that any max csp problem with a finite set of allowed constraint types which includes all fixed value constraints i e constraints of the form x e a is either solvable exactly in polynomial time or else is apx complete even if the number of occurrences of variables in instances is bounded moreover we present a simple description of all polynomial time solvable cases of our problem this description relies on the well known algebraic combinatorial property of supermodularity

on verlinde s formula for the dimensions of vector bundles on moduli spaces explicit expressions are given for the dimensions of vector bundles associated with the su 2 wess zumino witten model according to verlinde s formula

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

conformal field theory and the cohomology of the moduli space of stable bundles let g be a compact riemann surface of genus g 2 and let be a line bundle over g of degree 1 then the moduli space of rank 2 stable bundles v over g such that v was shown by seshadri 19 to be a nonsingular projective variety ng its rational cohomology ring h ng is exceedingly rich and despite years of study has never quite been computed in full the object of this paper is to give an essentially complete characterization of this ring or at least to reduce the problem to a matter of linear algebra in particular we find a proof of newstead s conjecture that pg1 ng 0 we also obtain a formula for the volume of ng which can be regarded as a twisted version of the formula for the degree 0 moduli space recently announced by witten the approach we shall take is not from algebraic geometry but from mathematical physics it relies on the su 2 wess zumino witten model which is a functor zk associating a finite dimensional vector space to each riemann surface with marked points the relationship with the moduli space is that when the level k of the functor is even the vector space associated to g with no marked points can be identified with h ng l where l is a fixed line bundle over ng now the work of verlinde provides us with a means of calculating the dimension of any vector space arising from our functor and in particular dimh ng l which we shall denote d g k on the other hand newstead 14 found explicit generators for h ng and we can also express d g k in terms of them using a riemann roch theorem equating the two formulas enables us to evaluate any monomial in the generators on the fundamental class of ng and by poincare duality this is sufficient at least in principle to determine the ring structure of h ng in the discussion above one important point has been skated over the su 2 wzw model is of course associated to bundles of degree 0 not degree 1 so in order to make use of verlinde s work it is necessary to formulate a twisted version of the field theory this is carried out in 2 but the crucial properties of the twisted theory analogous to those which make the ordinary theory a modular functor are not proved in this paper rather we will confine ourselves to exploring the consequences of these claims and hope to return to justify them in a later paper an outline of the remaining sections goes as follows in 3 we review those parts of verlinde s work we shall need show how they must be modified in the twisted case and work out some explicit formulas for d g k in 4 we study the cohomology of ng which we regard throughout as a space of representations via the theorem of narasimhan and seshadri we define newstead s generators and i of h ng

from local to global consistency abstract in reasoning tasks involving the maintenance of consistent databases so called qqconstraint networks q q it is customary to enforce local consistency conditions in order to simplify the subsequent construction of a globally coherent model of the data in this paper we present a relationship between the sizes of the variables domains the constraints arity and the level of local consistency sufficient to ensure global consistency based on these parameters a new tractability classification of constraint networks is presented we also show based on this relationship that any relation on bi valued variables which is not representable by a network of binary constraints cannot be represented by networks with any number of hidden variables

csp duality and trees of bounded pathwidth we study non uniform constraint satisfaction problems definable in monadic datalog stratified by the use of non linearity we show how such problems can be described in terms of homomorphism dualities involving trees of bounded pathwidth and in algebraic terms for this we introduce a new parameter for trees that closely approximates pathwidth and can be characterised via a hypergraph searching game