the complexity of constraint satisfaction revisited abstract this paper is a retrospective account of some of the developments leading up to and ensuing from the analysis of the complexity of some polynomial network consistency algorithms for constraint satisfaction problems

the expressive power of valued constraints hierarchies and collapses in this paper we investigate the ways in which a fixed collection of valued constraints can be combined to express other valued constraints we show that in some cases a large class of valued constraints of all possible arities can be expressed by using valued constraints over the same domain of a fixed finite arity we also show that some simple classes of valued constraints including the set of all monotonic valued constraints with finite cost values cannot be expressed by a subset of any fixed finite arity and hence form an infinite hierarchy

triangular norm based addition of fuzzy intervals triangular norm based addition of fuzzy intervals on the extended real line is reduced to the addition of shapes on 0 the addition of fuzzy intervals including lr fuzzy intervals lr fuzzy numbers lt and tr fuzzy numbers based on limit t norms is recalled for continuous archimedean t norms special types of incoming summands allowing the exact output computation are introduced several examples are given

t norm based addition of fuzzy intervals abstract the aim of this paper is to provide new results regarding the effective practical computation of t norm based additions of lr fuzzy intervals

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

domain permutation reduction for constraint satisfaction problems this paper is concerned with the constraint satisfaction problem csp it is well known that the general csp is np hard however there has been significant success in discovering subproblems which are tractable polynomial time solvable one of the most effective ways to obtain a tractable subproblem has been to force all of the constraint relations to lie in some tractable language in this paper we define a new way of identifying tractable subproblems of the csp let p be an arbitrary csp instance and c be any tractable language suppose there exists for each variable of p a permutation of the domain such the resultant permuted constraint relations of p all lie in c the domain permuted instance is then an instance of a tractable class and can be solved by the polynomial time algorithm for c solutions to p can be obtained by inverting the domain permutations the question for a given class of instances and language c whether such a set of domain permutations can be found efficiently is the key to this method s tractability one of the important contributions made in this paper is the notion of a lifted constraint instance which is a powerful tool to study this question we consider the open problem of discovering domain permutations which make instances max closed we show that for bounded arity instances over a boolean domain this problem is tractable while for domain size three it is intractable even for binary instances we give a simple proof verifying the tractability of discovering domain permutations which make instances row convex we refute a published result by giving a simple proof of the intractability of discovering domain permutations which make instances even with domain size four connected row convex we demonstrate that triangulated and stable marriage instances are reducible via domain permutations to max closed instances this provides a simple explanation for arc consistency deciding these instances we verify with a simple direct proof the tractability of identification of renamable horn instances and the intractability of finding the largest renamable horn subset of clauses of an instance of sat we describe natural tractable classes which properly extend the maximal relational classes arising from tractable constraint languages we believe that domain permutation reductions have a significant chance of being useful in practical applications

real vector space of lr fuzzy intervals with respect to the shape preserving t norm based addition the construction of the membership function of fuzzy intervals is an important problem in vagueness modeling the preservation of the shape of fuzzy sets during the addition is a natural requirement in practical computation the lr fuzzy intervals introduced by dubois and prade satisfy this requirement if the addition is based on the nilpotent t norm generated by l or r shape functions the shortcoming that not any lr fuzzy interval has an opposite inverse related to shape preserving t norm based addition can be solved if the set of lr fuzzy intervals is isomorphically included in an extended set and this extended set forms a group with respect to shape preserving t norm based addition in this paper we construct the extended set of these lr fuzzy intervals we also show that the extended set is a real vector space with scalar product and the modal intervals can be considered as the elements of this extended set finally we present the algebraic form of lr fuzzy intervals and the associated application

constraint relaxation may be perfect abstract networks of constraints are a simple knowledge representation method useful for describing those problems whose solution is required to satisfy several simultaneous constraints the problem of solving a network of constraints with finite domains is np complete the standard solution technique for such networks of constraints is the backtrack search but many relaxation algorithms to be applied before backtracking have been developed they transform a network in an equivalent but more explicit one the idea is to make the backtrack search have a better average time complexity in fact if the network elaborated by the backtrack algorithm is more explicit the algorithm backtracks less in this paper we describe relaxation algorithms as sequences of applications of relaxation rules moreover we define perfect relaxation algorithms as relaxation algorithms which not only return a more explicit network but also exactly solve the given network of constraints by applying every relaxation rule only once finally we characterize a family of classes of networks on which certain perfect relaxation algorithms are very efficient the exact solution of each network in a class is found in linear time

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

shape preserving additions of fuzzy intervals abstract triangular norm based additions preserving the lr shape of lr fuzzy intervals are studied the connection between t norm based additions preserving the linearity of fuzzy intervals and additions preserving lr shape for invertible shapes l and r is given a possibility of controlling the growth of the spread during the calculations with lr fuzzy intervals is shown

on the compositional rule of inference under triangular norms abstract the aim of this paper is to provide a close upper bound of the membership function for the compositional rule of inference under archimedean t norm where both the observation and the relation parts are given by hellendoorn s function 1980 in particular if the left and right spreads of the observation part is the same as those of the relation part then this upper bound is the exact membership function which generalizes the earlier result by fuller and zimmermann 1982 in that assumption of twice differentiability is deleted

learnability of quantified formulas we consider the following classes of quantified formulas fix a set of basic relations called a basis take conjunctions of these basic relations applied to variables and constants in arbitrary ways finally quantify existentially or universally some of the variables we introduce some conditions on the basis that guarantee efficient learnability furthermore we show that with certain restrictions on the basis the classification is complete we introduce as an intermediate tool a link between this class of quantified formulas and some well studied structures in universal algebra called clones more precisely we prove that the computational complexity of the learnability of these formulas is completely determined by a simple algebraic property of the basis of relations their clone of polymorphisms finally we use this technique to give a simpler proof of the already known dichotomy theorem over boolean domains

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

temporal constraint networks abstract this paper extends network based methods of constraint satisfaction to include continuous variables thus providing a framework for processing temporal constraints in this framework called temporal constraint satisfaction problem tcsp variables represent time points and temporal information is represented by a set of unary and binary constraints each specifying a set of permitted intervals the unique feature of this framework lies in permitting the processing of metric information namely assessments of time differences between events we present algorithms for performing the following reasoning tasks finding all feasible times that a given event can occur finding all possible relationships between two given events and generating one or more scenarios consistent with the information provided we distinguish between simple temporal problems stps and general temporal problems the former admitting at most one interval constraint on any pair of time points we show that the stp which subsumes the major part of vilain and kautz s point algebra can be solved in polynomial time for general tcsps we present a decomposition scheme that performs the three reasoning tasks considered and introduce a variety of techniques for improving its efficiency we also study the applicability of path consistency algorithms as preprocessing of temporal problems demonstrate their termination and bound their complexities

constraint tightness and looseness versus local and global consistency constraint networks are a simple representation and reasoning framework with diverse applications in this paper we identify two new complementary properties on the restrictiveness of the constraints in a network constraint tightness and constraint looseness and we show their usefulness for estimating the level of local consistency needed to ensure global consistency and for estimating the level of local consistency present in a network in particular we present a sufficient condition based on constraint tightness and the level of local consistency that guarantees that a solution can be found in a backtrack free manner the condition can be useful in applications where a knowledge base will be queried over and over and the preprocessing costs can be amortized over many queries we also present a sufficient condition for local consistency based on constraint looseness that is straightforward and inexpensive to determine the condition can be used to estimate the level of local consistency of a network this in turn can be used in deciding whether it would be useful to preprocess the network before a backtracking search and in deciding which local consistency conditions if any still need to be enforced if we want to ensure that a solution can be found in a backtrack free manner two definitions of local consistency are employed in characterizing the conditions the traditional variable based notion and a recently introduced definition of local consistency called relational consistency

on the minimality and global consistency of row convex constraint networks constraint networks have been shown to be useful in formulating such diverse problems as scene labeling natural language parsing and temporal reasoning given a constraint network we often wish to i find a solution that satisfies the constraints and ii find the corresponding minimal network where the constraints are as explicit as possible both tasks are known to be np complete in the general case task 1 is usually solved using a backtracking algorithm and task ii is often solved only approximately by enforcing various levels of local consistency in this paper we identify a property of binary constraint called row convexity and show its usefulness in deciding when a form of local consistency called path consistency is sufficient to guarantee that a network is both minimal and globally consistent globally consistent networks have the property that a solution can be found without backtracking we show that one can test for the row convexity property efficiently and we show by examining applications of constraint networks discussed in the literature that our results are useful in practice thus we identify a class of binary constraint networks for which we can solve both tasks i and ii efficiently finally we generalize the results for binary constraint networks to networks with nonbinary constraints

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

on equilibria of non compact generalized games abstract a fixed point theorem is first proved from which a theorem on the existence of maximal elements for lc majorized correspondences defined on a non compact set is obtained next an equilibrium theorem for non compact qualitative games with lc majorized correspondences defined on a non compact strategy set is proved and is applied to prove an equilibrium theorem for non compact generalized games with lc majorized correspondences defined on a non compact strategy set our results generalize several previously published corresponding results

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

t sum of l r fuzzy numbers abstract the sum of l r fuzzy numbers based on a given archimedian continuous t norm t is studied a series of improving sufficient conditions of several authors ensuring an analytical form of the output sum is completed by a sufficient and necessary condition i e the best sufficient condition some applications are indicated

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

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

tractable constraints on ordered domains abstract finding solutions to a constraint satisfaction problem is known to be an np complete problem in general but may be tractable in cases where either the set of allowed constraints or the graph structure is restricted in this paper we identify a restricted set of contraints which gives rise to a class of tractable problems this class generalizes the notion of a horn formula in propositional logic to larger domain sizes we give a polynomial time algorithm for solving such problems and prove that the class of problems generated by any larger set of constraints is np complete

characterising tractable constraints abstract finding solutions to a binary constraint satisfaction problem is known to be an np complete problem in general but may be tractable in cases where either the set of allowed constraints or the graph structure is restricted this paper considers restricted sets of constraints which are closed under permutation of the labels we identify a set of constraints which gives rise to a class of tractable problems and give polynomial time algorithms for solving such problems and for finding the equivalent minimal network we also prove that the class of problems generated by any set of constraints not contained in this restricted set is np complete

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

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

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

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

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

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

existentially restricted quantified constraint satisfaction the quantified constraint satisfaction problem qcsp is a framework for modelling pspace computational problems the general intractability of the qcsp has motivated the pursuit of restricted cases that avoid its maximal complexity in this paper we introduce and study a new model for investigating qcsp complexity in which the types of constraints given by the existentially quantified variables is restricted our primary technical contribution is the development and application of a general technology for proving positive results on parameterizations of the model of inclusion in the complexity class conp

on the convergence of t sum of l r fuzzy numbers abstract this paper presents the membership function of infinite or finite sum defined by the sup t norm convolution of l r fuzzy numbers under the conditions of the convexity of additive generators and the concavity of l r as an application we shall calculate the membership function of the limit distribution of the hamacher sum hr sum for 0 r 2 which generalizes fuller s results fuzzy sets and systems 41 1991 83 87 42 1991 205 212 in the case r 0 1 2

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

a sufficient condition for backtrack bounded search backtrack search is often used to solve constraint satisfaction problems a relationship involving the structure of the constraints is described that provides a bound on the backtracking required to advance deeper into the backtrack tree this analysis leads to upper bounds on the effort required for solution of a class of constraint satisfaction problems the solutions involve a combination of relaxation preprocessing and backtrack search the bounds are expressed in terms of the structure of the constraint connections specifically the effort is shown to have a bound exponential in the size of the largest biconnected component of the constraint graph as opposed to the size of the graph as a whole

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

a t sum bound of lr fuzzy numbers abstract in this paper we provide an upper bound and a lower bound of t sum of lr fuzzy numbers with different spreads where t is archimedean t norm and also show in three examples how close they are to actual membership functions furthermore we study when the membership function of t sum achieves the upper bound

on computation of the compositional rule of inference under triangular norms abstract this paper is devoted to the derivation of exact calculation formulas for the compositional rule of inference under archimedean t norms when both the observation and the relation parts are given by hellendoorn s function 6

generalized vector quasi equilibrium problems with applications in this paper we consider the generalized vector quasi equilibrium problem with or without involving condensing maps and prove the existence of its solution by using known fixed point and maximal element theorems as applications of our results we derive some existence results for a solution to the vector quasi optimization problem for nondifferentiable functions and vector quasi saddle point problem

decomposing constraint satisfaction problems using database techniques abstract there is a very close relationship between constraint satisfaction problems and the satisfaction of join dependencies in a relational database which is due to a common underlying structure namely a hypergraph by making that relationship explicit we are able to adapt techniques previously developed for the study of relational databases to obtain new results for constraint satisfaction problems in particular we prove that a constraint satisfaction problem may be decomposed into a number of subproblems precisely when the corresponding hypergraph satisfies a simple condition we show that combining this decomposition approach with existing algorithms can lead to a significant improvement in efficiency

on the approximation of upper semi continuous correspondences and the equilibriums of generalized games in section 1 we explain some of the definitions and terminology that we use in section 2 we prove several theorems concerning the approximation of upper semi continuous correspondences having for range a locally convex space theorems 1 and 2 and corollary 1 generalize certain approximation theorems by g haddad 16 pp 1352213541 g haddad and j m lasry 17 pp 299 3001 and j p aubin and a cellina 2 pp 86891 see also f s de blasi lo and j m lasry and r robert 21 although some of the details of the proofs of theorems 1 and 2 are new the basic ideas are taken from the above mentioned papers theorem 3 shows that the correspondences in the approximating families can be chosen so that they are regular see section 1 this theorem see the remark following its proof contains a classical result of m hukuhara 18 pp 56573 theorem 3 is proved using theorem 2 and propositions 1 and 4 variants of propositions 2 3 and 4 were given in 193 in section 3 we give among others theorem 5 which concerns the existence of equilibriums of generalized games abstract economies the main purpose of this result is to replace the continuity hypothesis in the w shafer h sonnenschein equilibrium theorem for generalized games by an upper semi continuity one the proof of theorem 5 is based on the results on the approximation of upper semi continuous correspondences obtained in section 2 theorem 5 us used in section 4 in section 4 we establish theorems 6 and 7 these theorems show that certain statements concerning the equilibrium of generalized games are equivalent to certain statements concerning minimax inequalities of k fan type theorems 1 2 3 5 6 and 7 are the main results of this paper 267 0022 247x 88 3 00

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

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

constraint propogation techniques for the disjunctive scheduling problem abstract constraint propagation is an elementary method for reducing the search space of combinatorial search and optimization problems which has become more and more important in the last decades the basic idea of constraint propagation is to detect and remove inconsistent variable assignments that cannot participate in any feasible solution through the repeated analysis and evaluation of the variables domains and constraints describing a specific problem instance the contribution of this paper is twofold the first contribution is a description of efficient constraint propagation methods also known as consistency tests for the disjunctive scheduling problem dsp which is a generalization of the classical job shop scheduling problem jsp applying an elementary constraint based approach involving a limited number of search variables we will derive consistency tests that ensure 3 b consistency we will further present and analyze both new and classical consistency tests which to some extent are generalizations of the aforementioned consistency tests involving a higher number of variables but still can be implemented efficiently with a polynomial time complexity further the concepts of energetic reasoning and shaving are analyzed and discussed the other contribution is a classification of the consistency tests derived according to the domain reduction achieved the particular strength of using consistency tests is based on their repeated application so that the knowledge derived is propagated i e reused for acquiring additional knowledge the deduction of this knowledge can be described as the computation of a fixed point since this fixed point depends upon the order of the application of the tests we first derive a necessary condition for its uniqueness we then develop a concept of dominance which enables the comparison of different consistency tests as well as a simple method for proving dominance an extensive comparison of all consistency tests is given quite surprisingly we will find out that some apparently stronger consistency tests are subsumed by apparently weaker ones at the same time an open question regarding the effectiveness of energetic reasoning is answered

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

a note to the t sum of l r fuzzy numbers abstract we generalize the results of hong and hwang 1994 for the membership function of finite infinite sum of l r fuzzy numbers where the sum is based on some continuous archimedean t norm t

constant query testability of assignments to constraint satisfaction problems for each finite relational structure a let csp a denote the csp instances whose constraint relations are taken from a the resulting family of problems csp a has been considered heavily

properties of tree convex constraints it is known that a tree convex network is globally consistent if it is path consistent however if a tree convex network is not path consistent enforcing path consistency on it may not make it globally consistent in this paper we investigate the properties of some tree convex constraints under intersection and composition as a result we identify a sub class of tree convex networks that are locally chain convex and strictly union closed this class of problems can be made globally consistent by arc and path consistency and thus is tractable interestingly we also find that some scene labeling problems can be modeled by tree convex constraints in a natural and meaningful way

a further generalization of yannelis prabhakar s continuous selection theorem and its applications abstract in this article we first prove one improved version of the yannelis prabhakar continuous selection theorem and next as its applications a fixed point theorem in noncompact product spaces a nonempty intersection theorem some existence theorems of solutions for the generalized quasi variational inequalities and some equilibrium existence theorems for the abstract economies are given

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

collective fixed points and maximal elements with applications to abstract economies abstract in this paper we first establish collective fixed points theorems for a family of multivalued maps with or without assuming that the product of these multivalued maps is condensing as an application of our collective fixed points theorem we derive the coincidence theorem for two families of multivalued maps defined on product spaces then we give some existence results for maximal elements for a family of ls majorized multivalued maps whose product is condensing we also prove some existence results for maximal elements for a family of multivalued maps which are not ls majorized but their product is condensing as applications of our results some existence results for equilibria of abstract economies are also derived the results of this paper are more general than those given in the literature

building tractable disjunctive constraints many combinatorial search problems can be expressed as constraint satisfaction problems this class of problems is known to be np hard in general but a number of restricted constraint classes have been identified which ensure tractability this paper presents the first general results on combining tractable constraint classes to obtain larger more general tractable classes we give examples to show that many known examples of tractable constraint classes from a wide variety of different contexts can be constructed from simpler tractable classes using a general method we also construct several new tractable classes that have not previously been identified

a simple algorithm for mal tsev constraints a mal tsev operation is a ternary operation varphi that satisfies the identities varphi x y y varphi y y x x constraint satisfaction problems involving constraints invariant under a mal tsev operation constitute an important class of constraint satisfaction problems which includes the affine satisfiability problem subgroup and near subgroup constraints and many others it is also known that any tractable case of the counting constraint satisfaction problem involves only mal tsev constraints the first algorithm solving the arbitrary constraint satisfaction problem with mal tsev constraints has been given by bulatov however this algorithm is very sophisticated and relies heavily on advanced algebraic machinery in this paper we give a different and much simpler algorithm for this type of constraint

on binary constraint problems the concepts of binary constraint satisfaction problems can be naturally generalized to the relation algebras of tarski the concept of path consistency plays a central role algorithms for path consistency can be implemented on matrices of relations and on matrices of elements from a relation algebra we give an example of a 4 by 4 matrix of infinite relations on which on iterative local path consistency algorithm terminates we give a class of examples over a fixed finite algebra on which all iterative local algorithms whether parallel or sequential must take quadratic time specific relation algebras arising from interval constraint problems are also studied the interval algebra the point algebra and the containment algebra

equilibria of non compact generalized games with l majorized preference correspondences an existence theorem of maximal elements in a non compact set for l majorized correspondences is obtained next an existence theorem of an equilibrium in a qualitative game is proved and is then applied to achieve an existence theorem of equilibrium in a non compact abstract economy with l majorized preference correspondences these results are either very closely related to or generalizations of those recent results of borglin keiding yannelis prabhakar toussaint and tulcea

fixed point and maximal element theorems with applications to abstract economies and minimax inequalities in this paper we establish some fixed point theorems for a family of multivalued maps under mild conditions by using our fixed point theorems we derive some maximal element theorems for a particular family of multivalued maps namely the condensing multivalued maps as applications of our results we prove some general equilibrium existence theorems in the generalized abstract economies with preference correspondences further applications of our results are also given to minimax inequalities for a family of functions

additions of interactive fuzzy numbers this paper provides an account of an approach to modeling unknown data by means of fuzzy set theory and addresses the problem of deriving the uncertainty on a sum of variables whose values lie within fuzzy intervals the first part is an extensive presentation of the theoretical background of the approach the extension principle is stated in terms of possibility of an event the concept of variable interaction is investigated at length section ii gives new results regarding the effective practical computation of additions of fuzzy numbers its originality lies in the introduction of interaction which enables to control the growth of uncertainty in calculations moreover the problem of computing mathematical expectations with fuzzy probabilities is solved the results derived in this paper can easily be used in decision problems where values of parameters or decision variables are not yet precisely fixed or assessed typical applications could be multicriteria optimization and decision making under uncertainty where fuzzy expected utilities can be obtained out of uncompletely assessed probabilities more generally fuzzy arithmetic can be an important tool for sophisticated computationally tractable sensitivity analysis in systems modeling computer aided design and operations research

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

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

fast parallel constraint satisfaction a constraint satisfaction problem csp involves searching for an assignment of values to a given set of variables so that the values assigned satisfy a given set of constraints the general csp is np complete to confront the intractability of the general csp relaxation procedures have been devised instead of searching for a globally consistent assignment of values to the variables try to restrict the domain of values of each variable in a way that ensures local consistency only the relaxation procedures are efficient but have been proved to be inherently sequential in this paper we define a class of csps for which a global solution can be found by a fast parallel algorithm no relaxation preprocessing is needed for the parallel algorithm to work the result is motivated from the problem of labelling a 2 d line drawing of a 3 d object by the clowes huffman malik labelling scheme an important application of csp in computer vision for such a labelling csp the constraint graph can be general but the constraint relations are usually of the type we call implicational it is shown here that a csp with this type of constraint relations and no restrictions on its graph can be solved by an efficient i e with polynomial time complexity sequential algorithm also it is shown that it can be solved by a fast parallel algorithm that executes in time o log3n with o m n3 log n processors on a exlusive read exclusive write parallel random access machine n is the number of variables and m is the number of constrain relations the constraint relations may have arity more than two

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

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

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

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

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

nilpotent t norm based sum of fuzzy intervals in this paper the sum of l r fuzzy intervals based on a given nilpotent triangular norm t norm t is studied by modifying markova s results which gives the necessary and sufficient condition ensuring an analytical form of the output sum to the form with arbitrary spreads we generalize the results of hong and hwang for the membership function of the sum of l r fuzzy intervals where the sum is based on some archimedean t norm t

half integrality lp branching and fpt algorithms a recent trend in parameterized algorithms is the application of polytope tools to fixed parameter tractable fpt algorithms e g cygan et al focs 2011 52nd annual symposium on foundations of computer science ieee 2011 pp 150 159 narayanaswamy et al stacs 2012 symposium on theoretical aspects of computer science 2012 pp 338 349 although this approach has yielded significant speedups for a range of important problems it requires the underlying polytope to have very restrictive properties including half integrality and nemhauser trotter style persistence properties to date these properties are essentially known to hold only for two classes of polytopes covering the cases of vertex cover nemhauser and trotter math program 8 1975 pp 232 248 and node multiway cut garg et al j alg 50 2004 pp 49 61 taking a slightly different approach we view half integrality as a discrete relaxation of a problem e g a relaxation of the search space from 0 1 v to 0

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