reducts of the random partial order abstract we determine up to the equivalence of first order interdefinability all structures which are first order definable in the random partial order it turns out that there are precisely five such structures we achieve this result by showing that there exist exactly five closed permutation groups which contain the automorphism group of the random partial order and thus expose all symmetries of this structure our result lines up with previous similar classifications for the random graph and the order of the rationals it also provides further evidence for a conjecture due to simon thomas which states that the number of structures definable in a homogeneous structure in a finite relational language is up to first order interdefinability always finite in the proof we use the new technique of canonical functions originally invented in the context of theoretical computer science which allows for a systematic ramsey theoretic analysis of functions acting on the random partial order the technique identifies patterns in arbitrary functions on the random partial order which makes them accessible to finite combinatorial arguments

discrete painleve equations coalescences limits and degeneracies abstract starting from the standard form of the five discrete painleve equations we show how one can obtain through appropriate limits a host of new equations which are also the discrete analogues of the continuous painleve equations a particularly interesting technique is the one based on the assumption that some simplification takes place in the autonomous form of the mapping following which the deautonomization leads to a new n dependence and introduces more new discrete painleve equations

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

constants and finite unary relations in qualitative constraint reasoning extending qualitative csps with the ability of restricting selected variables to finite sets of possible values has been proposed as an interesting research direction with important applications c

a sensitivity tradeoff for plants with time delay in this paper we show that the sensitivity function of a scalar feedback system must satisfy an integral constraint when the open loop transfer function is strictly proper and contains a time delay the integral constraint is identical to the classical bode sensitivity integral which holds provided the open loop gain has greater than a one pole rolloff the design implications of the two constraints are somewhat different however and this difference is discussed the relation between open loop phase and a conflict between bandwidth and sensitivity properties is explored

strongly asymmetric discrete painleve equations the additive case we examine a class of discrete painleve equations which present a strong asymmetry these equations can be written as a system of two equations the right hand sides of which do not have the same functional form we limit here our investigation to two canonical families of the quispel roberts thompson qrt classification both of which lead to difference equations several new integrable discrete systems are identified

an affine weyl group approach to the eight parameter discrete painleve equation we present a geometrical construction of the eight parameter discrete painleve equations our starting point is the e 1 8 affine weyl group we assume that the multi dimensional function lives on the vertices of the weight lattice of this group we derive the bilinear equations related to the discrete painleve equation in the form of nonautonomous hirota miwa equations and the elementary miura transformations the compatibility condition of the various miura s that can be written leads to three types of equations difference multiplicative q and another type where the parameters and the independent variable enter through the arguments of elliptic functions we write explicitly the discrete equations in the first two cases and produce their degeneration through coalescence of parameters

limits and degeneracies of discrete painleve equations a sequel abstract we present the discrete systems which result from the discrete painleve equations q p vi and d p v associated to the affine weyl group e 7 1 two different procedures limits and degeneracies are used giving rise to a host of new discrete painleve equations but also to some equations which are integrable through linearisation

logarithmic integrals interpolation bounds and performance limitations in mimo feedback systems we study performance limitation issues found in linear multivariable feedback systems our main contributions include bode and poisson type integral inequalities and performance limits for the sensitivity and complementary sensitivity functions these results characterize and quantify explicitly how open loop unstable poles and nonminimum phase zeros may impose inherent limitations on feedback design and fundamental limits on the best achievable performance the role of time delay is also studied in this context most notably we show that the performance and design limitations in multivariable systems intrinsically depend on the locations as well as directions of unstable poles and nonminimum phase zeros and in particular on how pole and zero directions are aligned the latter is characterized by angles measuring the mutual orientation between zero and pole directions and it is shown to play a crucial role in multivariable system design

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

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

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

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

feedback minimax sensitivity and optimal robustness in this paper we look for feedbacks that minimize the sensitivity function of a linear single variable feedback system represented by its frequency responses sensitivity to disturbances and robustness under plant perturbations are measured in a weighted h infty norm in an earlier paper zames proposed an approach to feedback design involving the measurement of sensitivity by multiplicative seminorms which have certain advantages over the widely used quadratic norm in problems where there is plant uncertainty or where signal power spectra are not fixed but belong to sets the problem was studied in a general setting and some h infty examples were solved here a detailed study of the single variable case is undertaken the results are extended to unstable plants and explicit formulas for the general situation of a finite number of right half plane rhp plant zeros or poles are provided the q or approximate inverse parametrization of feedbacks that maintain closed loop stability is extended to the ease of unstable plants the h infty and wiener hopf approaches are compared

point algebras for temporal reasoning algorithms and complexity we investigate the computational complexity of temporal reasoning in different time models such as totally ordered partially ordered and branching time our main result concerns the satisfiability problem for point algebras and point algebras extended with disjunctions for these problems we identify all tractable subclasses we also provide a number of additional results for instance we present a new time model suitable for reasoning about systems with a bounded number of unsynchronized clocks we investigate connections with spatial reasoning and we present improved algorithms for deciding satisfiability of the tractable point algebras

on a novel representation of discrete painleve equations we introduce a new representation for the discrete painleve equations it is based on the trihomographic form previously used in conjunction with discrete painleve equations associated with the affine weyl group e8 1 we show here that all discrete painleve equations do possess a trihomographic form the latter is often more convenient than the usual representation in particular when it comes to the application of the singularity confinement integrability criterion we show how trihomographic forms can be combined in order to construct generic discrete painleve equations or limits thereof a trihomographic form for the generic elliptic painleve equation is also derived

restoring discrete painleve equations from an e8 1 associated one we present a systematic method for the construction of discrete painleve equations the method dubbed restoration allows one to obtain all discrete painleve equations that share a common autonomous limit up to homographic transformations starting from any one of those limits as the restoration process crucially depends on the classification of canonical forms for the mappings in the quispel roberts thompson qrt family it can in principle only be applied to mappings that belong to that family however as we show in this paper it is still possible to obtain the results of the restoration even when the initial mapping is not of the qrt type at least for the system at hand but we believe our approach to be of much wider applicability for the equations derived in this paper we also show how starting from a form where the independent variable advances one step at a time one can obtain versions corresponding to multistep evolutions we present a systematic method for the construction of discrete painleve equations the method dubbed restoration allows one to obtain all discrete painleve equations that share a common autonomous limit up to homographic transformations starting from any one of those limits as the restoration process crucially depends on the classification of canonical forms for the mappings in the quispel roberts thompson qrt family it can in principle only be applied to mappings that belong to that family however as we show in this paper it is still possible to obtain the results of the restoration even when the initial mapping is not of the qrt type at least for the system at hand but we believe our approach to be of much wider applicability for the equations derived in this paper we also show how starting from a form where the independent variable advances one step at a time one can obtain versions corresponding to multistep evolutions

scheduling with and or precedence constraints in many scheduling applications it is required that the processing of some job be postponed until some other job which can be chosen from a pregiven set of alternatives has been completed the traditional concept of precedence constraints fails to model such restrictions therefore the concept has been generalized to so called and or precedence constraints which can cope with this kind of requirement in the context of traditional precedence constraints feasibility transitivity and the computation of earliest start times for jobs are fundamental well studied problems the purpose of this paper is to provide efficient algorithms for these tasks for the more general model of and or precedence constraints we show that feasibility as well as many questions related to transitivity can be solved by applying essentially the same linear time algorithm in order to compute earliest start times we propose two polynomial time algorithms to cope with different classes of time distances between jobs

an optimal algorithm for approximate nearest neighbor searching fixed dimensions consider a set of s of n data points in real d dimensional space r d where distances are measured using any minkowski metric in nearest neighbor searching we preprocess s into a data structure so that given any query point q r d is the closest point of s to q can be reported quickly given any positive real e data point p is a 1 e approximate nearest neighbor of q if its distance from q is within a factor of 1 e of the distance to the true nearest neighbor we show that it is possible to preprocess a set of n points in r d in o dn log n time and o dn space so that given a query point q r d and e 0 a 1 e approximate nearest neighbor of q can be computed in o c d e log n time where c d e d 1 6d e d is a factor depending only on dimension and e in general we show that given an integer k 1 1 e approximations to the k nearest neighbors of q can be computed in additional o kd log n time

shorter arithmetization of nondeterministic computations arithmetizing computation is a crucial component of many fundamental results in complexity theory including results that gave insight into the power of interactive proofs multi prover interactive proofs and probabilistically checkable proofs informally an arithmetization is a way to encode a machine s computation so that its correctness can be easily verified via few probabilistic algebraic checks we study the problem of arithmetizing nondeterministic computations for the purpose of constructing short probabilistically checkable proofs pcps with polylogarithmic query complexity in such a setting a pcp s proof length depends at least linearly on the length in bits of the encoded computation thus minimizing the number of bits in the encoding is crucial for minimizing pcp proof length in this paper we show how to arithmetize any t step computation on a nondeterministic turing machine by using a polynomial encoding of length o t log t 2 previously the best known length was t log t 4 for nondeterministic random access machines our length is o t log t 2 o 1 while prior work only achieved t log t 5 the polynomial encoding that we use is the reed solomon code when combined with the best pcps of proximity for this code our result yields quasilinear size pcps with polylogarithmic query complexity that are shorter by at least two logarithmic factors than in all prior work our arithmetization also enjoys additional properties first it is succinct i e the encoding of the computation can be probabilistically checked in log t o 1 time this property is necessary for constructing short pcps with a polylogarithmic time verifier furthermore our techniques extend in a certain well defined sense to the arithmetization of yet other nexp complete languages

twenty one large tractable subclasses of allen s algebra abstract this paper continues nebel and burckert s investigation of allen s interval algebra by presenting nine more maximal tractable subclasses of the algebra provided that p np in addition to their previously reported ord horn subclass furthermore twelve tractable subclasses are identified whose maximality is not decided four of them can express the notion of sequentiality between intervals which is not possible in the ord horn algebra all of the algebras are considerably larger than the ord horn subclass the satisfiability algorithm which is common for all the algebras is shown to be linear furthermore the path consistency algorithm is shown to decide satisfiability of interval networks using any of the algebras

integrable mappings and soliton equations ii abstract some simple solutions of discrete soliton equations are shown to satisfy 2d mappings we show that these belong to a recently introduced 18 parameter family of integrable reversible mappings of the plane thus lending weight to a previous conjecture we also give an example of an integrable mapping occuring in an exactly solvable model in statistical mechanics finally we discuss the notion of generalized reversibility

paper sensitivity integrals for multivariable discrete time systems the purpose of this paper is to develop integral formulae for singular values of the sensitivity function to express design constraints in multivariable discrete time systems we present extensions to both the classical poisson integral and bode s sensitivity integral formulae the main utility of these results is that they can be used to quantify design limitations that arise in multivariable discrete time systems due to such system characteristics as open loop unstable poles and nonminimum phase zeros and to such design requirements as stability and bandwidth constraints these formulae are similar to those for multivariable continuous time systems obtained elsewhere and they reveal that feedback design limitations depend on directionality properties of the sensitivity function and on those of unstable poles and nonminimum phase zeros in the open loop transfer function

discretising the painleve equations a la hirota mickens we present a systematic method for discretising the painleve equations inspired by the method of hirota while extending it and by that of mickens by specifying it to the case at hand we derive various discrete analogues of painleve i and ii we obtain forms that have been previously derived as well as new ones in particular equations with a geometry described by the affine weyl group e8 1 as a by product we obtain also linearisable equations some of which are new

biased search trees we consider the problem of storing items from a totally ordered set in a search tree so that the access time for a given item depends on a known estimate of the access frequency of the item we describe two related classes of biased search trees whose average access time is within a constant factor of the minimum and that are easy to update under insertions deletions and more radical update operations we present and analyze efficient update algorithms for biased search trees we list several applications of such trees

fast construction of nets in low dimensional metrics and their applications we present a near linear time algorithm for constructing hierarchical nets in finite metric spaces with constant doubling dimension this data structure is then applied to obtain improved algorithms for the following problems approximate nearest neighbor search well separated pair decomposition spanner construction compact representation scheme doubling measure and computation of the approximate lipschitz constant of a function in all cases the running preprocessing time is near linear and the space being used is linear

integrable mappings of the plane preserving biquadratic invariant curves we provide a general framework to construct integrable mappings of the plane that preserve a one parameter family b x y k of biquadratic invariant curves where parametrization by k is very general these mappings are reversible by construction i e they are the composition of two involutions and can be shown to be measure preserving they generalize integrable maps previously given by mcmillan and quispel roberts and thompson by considering a transformation of the case of the symmetric biquadratic to a canonical form we provide a normal form for the symmetric integrable map acting on each invariant curve we give a lax pair for a large subclass of our symmetric integrable maps including at least a 10 parameter subfamily of the 12 parameter symmetric quispel roberts thompson maps

space time tradeoffs for approximate nearest neighbor searching nearest neighbor searching is the problem of preprocessing a set of n point points in d dimensional space so that given any query point q it is possible to report the closest point to q rapidly in approximate nearest neighbor searching a parameter e 0 is given and a multiplicative error of 1 e is allowed we assume that the dimension d is a constant and treat n and e as asymptotic quantities numerous solutions have been proposed ranging from low space solutions having space o n and query time o log n 1 ed 1 to high space solutions having space roughly o n log n ed and query time o log n e we show that there is a single approach to this fundamental problem which both improves upon existing results and spans the spectrum of space time tradeoffs given a tradeoff parameter where 2 1 e we show that there exists a data structure of space o n d 1 log 1 e that can answer queries in time o log n 1 e d 1 2 when e 2 this yields a data structure of space o n log 1 e that can answer queries in time o log n 1 e d 1 2 when e 1 e it provides a data structure of space o n ed 1 log 1 e that can answer queries in time o log n e our results are based on a data structure called a t e avd which is a hierarchical quadtree based subdivision of space into cells each cell stores up to t representative points of the set such that for any query point q in the cell at least one of these points is an approximate nearest neighbor of q we provide new algorithms for constructing avds and tools for analyzing their total space requirements we also establish lower bounds on the space complexity of avds and show that up to a factor of o log 1 e our space bounds are asymptotically tight in the two extremes e 2 and e 1 e

maximal infinite valued constraint languages we systematically investigate the computational complexity of constraint satisfaction problems for constraint languages over an infinite domain in particular we study a generalization of the well established notion of maximal constraint languages from finite to infinite domains if the constraint language can be defined with an w categorical structure then maximal constraint languages are in one to one correspondence to minimal oligomorphic clones based on this correspondence we derive general tractability and hardness criteria for the corresponding constraint satisfaction problems

satisfying ternary permutation constraints by multiple linear orders or phylogenetic trees a ternary permutation constraint satisfaction problem csp is specified by a subset of the symmetric group s 3 an instance of such a problem consists of a set of variables v and a set of constraints c where each constraint is an ordered triple of distinct elements from v the goal is to construct a linear ordering on v such that for each constraint a b c c the ordering of a b c induced by is in excluding symmetries and trivial cases there are eleven such problems and their complexity is well known here we consider the variant of the problem denoted 2 where we are allowed to construct two linear orders and and each constraint needs to be satisfied by at least one of the two we give a full complexity classification of all eleven 2 problems observing that in the switch from one to two linear orders the complexity landscape changes quite abruptly and that hardness proofs become rather intricate to establish the proofs we use several computer aided searches each such search returns a small instance with a unique solution for a given problem we then focus on one of the eleven problems in particular which is closely related to the 2 caterpillar compatibility problem in the phylogenetics literature we show that this particular csp remains hard on three linear orders and also in the biologically relevant case when we swap three linear orders for three phylogenetic trees yielding the 3 tree compatibility problem lastly we give extremal results concerning the minimum number of trees required in the worst case to satisfy a set of rooted triplet constraints on n leaf labels

crossover frequency limitations in mimo nonminimum phase feedback systems this paper investigates limitations and design tradeoffs of the closed loop sensitivity performance of linear time invariant nonminimum phase uncertain multiple input multiple output plants with i inputs and m outputs where m spl les l it is shown that if rows i sub 1 i sub k of the plant transfer function form a k spl times l nonminimum phase transfer matrix and if the design is such that the sensitivity gain of k 1 rows among the rows i sub 1 i sub k of the closed loop transfer function is low then by necessity the sensitivity gain of the remaining row is high this sensitivity constraint is quantified with the help of the crossover frequency restriction of a specially constructed single input single output transfer function that includes the right half plane zeros and poles of the k spl times l transfer matrix

feedback and optimal sensitivity model reference transformations multiplicative seminorms and approximate inverses in this paper the problem of sensitivity reduction by feedback is formulated as an optimization problem and separated from the problem of stabilization stable feedback schemes obtainable from a given plant are parameterized salient properties of sensitivity reducing schemes are derived and it is shown that plant uncertainty reduces the ability of feedback to reduce sensitivity the theory is developed for input output systems in a general setting of banach algebras and then specialized to a class of multivariable time invariant systems characterized by n times n matrices of h infty frequency response functions either with or without zeros in the right half plane the approach is based on the use of a weighted seminorm on the algebra of operators to measure sensitivity and on the concept of an approximate inverse approximate invertibility of the plant is shown to be a necessary and sufficient condition for sensitivity reduction an indicator of approximate invertibility called a measure of singularity is introduced the measure of singularity of a linear time invariant plant is shown to be determined by the location of its right half plane zeros in the absence of plant uncertainty the sensitivity to output disturbances can be reduced to an optimal value approaching the singularity measure in particular if there are no right half plane zeros sensitivity can be made arbitrarily small the feedback schemes used in the optimization of sensitivity resemble the lead lag networks of classical control design some of their properties and methods of constructing them in special cases are presented

sensitivity integrals and transformation techniques a new perspective we provide alternative proofs for the classical bode and poisson type sensitivity integrals and their extensions for both continuous time and discrete time systems our derivation uses the well known properties of laplace and z transformations this derivation helps establish a connection between bode and poisson type sensitivity integrals and laplace and z transforms hence providing an alternative perspective in interpreting these fundamental integral results

an initial study of time complexity in infinite domain constraint satisfaction abstract the constraint satisfaction problem csp is a widely studied problem with numerous applications in computer science and artificial intelligence for infinite domain csps there are many results separating tractable and np hard cases while upper and lower bounds on the time complexity of hard cases are virtually unexplored hence we initiate a study of the worst case time complexity of such csps we analyze backtracking algorithms and determine upper bounds on their time complexity we present asymptotically faster algorithms based on enumeration techniques and we show that these algorithms are applicable to well studied problems in for instance temporal reasoning finally we prove non trivial lower bounds applicable to many interesting csps under the assumption that certain complexity theoretic assumptions hold the gap between upper and lower bounds is in many cases surprisingly small which suggests that our upper bounds cannot be significantly improved

multivariable gain phase and sensitivity integral relations and design trade offs we present several extensions of the classical bode integral relations to multivariable systems the main contributions are an extended gain phase integral formula and an extended sensitivity integral relation these results are useful for analyzing fundamental design limitations that arise in linear multivariable feedback systems in particular they provide important insights into feedback properties which are unique to multivariable systems and which cannot be inferred from the previously known integral relations applicable to single input single output systems both results show that design limitations in multivariable systems depend strongly upon certain directionality properties

reasoning about temporal relations the tractable subalgebras of allen s interval algebra allen s interval algebra is one of the best established formalisms for temporal reasoning this article provides the final step in the classification of complexity for satisfiability problems over constraints expressed in this algebra when the constraints are chosen from the full allen s algebra this form of satisfiability problem is known to be np complete however eighteen tractable subalgebras have previously been identified we show here that these subalgebras include all possible tractable subsets of allen s algebra in other words we show that this algebra contains exactly eighteen maximal tractable subalgebras and reasoning in any fragment not entirely contained in one of these subalgebras is np complete we obtain this dichotomy result by giving a new uniform description of the known maximal tractable subalgebras and then systematically using a general algebraic technique for identifying maximal subalgebras with a given property

investigating relations between discrete painleve equations the multistep approach we show how starting from a mapping where the independent variable advances one step at a time one can obtain versions of the mapping corresponding to a multi step evolution the same procedure is applied to discrete painleve equations and we proceed to establish miura relations between the single step and the multi step versions in the present study multi referring to double triple and quintuple these miura relations are discrete painleve equations on their own right we show that while in some cases it is impossible to obtain a multi step equation for a single variable deriving a miura system is still possible we perform our analysis for equations associated with the affine weyl groups e8 1 e7 1 e6 1 and a4 1

integral constraints and performance limits on complementary sensitivity discrete time systems abstract we study performance limitation issues for multivariable discrete time feedback systems the complementary sensitivity function is employed as a performance measure and bode and poisson type integral inequalities and h type performance limits are derived the results exhibit frequency dependent constraints as well as best achievable limits on the complementary sensitivity function which are shown to be determined by nonminimum phase zeros unstable poles and time delays in particular the directions of such zeros and poles are seen to play a central role to this effect

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

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 decomposition of multidimensional point sets with applications to k nearest neighbors and n body potential fields we define the notion of a well separated pair decomposition of points in d dimensional space we then develop efficient sequential and parallel algorithms for computing such a decomposition we apply the resulting decomposition to the efficient computation of k nearest neighbors and n body potential fields

logarithmic integrals and system dynamics an analogue of bode s sensitivity integral for continuous time time varying systems a new time domain interpretation of bode s integral is presented this allows for a generalization to the class of time varying systems which possess an exponential dichotomy it is shown that the sensitivity function is constrained on average by the spectral values in the dichotomy spectrum of the antistable component of the open loop dynamics

a new class of integrable discrete systems we derive a new class of integrable autonomous mappings which possess invariants biquartic in the dependent variables we link these mappings to discrete painleve equations we show how mappings with biquartic invariants can be naturally built up from the requirement that their solutions be parametrized in terms of elliptic functions

right half plane poles and zeros and design tradeoffs in feedback systems this paper expresses limitations imposed by right half plane poles and zeros of the open loop system directly in terms of the sensitivity and complementary sensitivity functions of the closed loop system the limitations are determined by integral relationships which must be satisfied by these functions the integral relationships are interpreted in the context of feedback design

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

schaefer s theorem for graphs schaefer s theorem is a complexity classification result for so called boolean constraint satisfaction problems it states that every boolean constraint satisfaction problem is either contained in one out of six classes and can be solved in polynomial time or is np complete we present an analog of this dichotomy result for the propositional logic of graphs instead of boolean logic in this generalization of schaefer s result the input consists of a set w of variables and a conjunction of statements constraints about these variables in the language of graphs where each statement is taken from a fixed finite set of allowed quantifier free first order formulas the question is whether is satisfiable in a graph we prove that either is contained in one out of 17 classes of graph formulas and the corresponding problem can be solved in polynomial time or the problem is np complete this is achieved by a universal algebraic approach which in turn allows us to use structural ramsey theory to apply the universal algebraic approach we formulate the computational problems under consideration as constraint satisfaction problems csps whose templates are first order definable in the countably infinite random graph our method for classifying the computational complexity of those csps is based on a ramsey theoretic analysis of functions acting on the random graph and we develop general tools suitable for such an analysis which are of independent mathematical interest

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

reasoning about qualitative temporal information abstract representing and reasoning about incomplete and indefinite qualitative temporal information is an essential part of many artificial intelligence tasks an interval based framework and a point based framework have been proposed for representing such temporal information in this paper we address two fundamental reasoning tasks that arise in applications of these frameworks given possibly indefinite and incomplete knowledge of the relationships between some intervals or points i find a scenario that is consistent with the information provided and ii find the feasible relations between all pairs of intervals or points for the point based framework and a restricted version of the interval based framework we give computationally efficient procedures for finding a consistent scenario and for finding the feasible relations our algorithms are marked improvements over the previously known algorithms in particular we develop an o n2 time algorithm for finding one consistent scenario that is an o n improvement over the previously known algorithm where n is the number of intervals or points and we develop an algorithm for finding all the feasible relations that is of far more practical use than the previously known algorithm for the unrestricted version of the interval based framework finding a consistent scenario and finding the feasible relations have been shown to be np complete we show how the results for the point algebra aid in the design of a backtracking algorithm for finding one consistent scenario that is shown to be useful in practice for planning problems

from trihomographic to elliptic painleve equations we use the trihomographic representation of discrete painleve equations in order to derive a new factorised form for equations associated with the affine weyl group introducing an ancillary dependent variable we are able to obtain factorised forms for the general additive multiplicative and elliptic discrete related painleve equations the advantage of these factorised forms is that they make the singularity analysis of the straightforward in par with the situation for equations associated with lower affine weyl groups

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

on the autonomous limit of discrete painleve equations abstract we consider the autonomous limit of the known families of discrete painleve equations we show how these autonomous systems can be written as special cases of the qrt mapping for the latter we present in detail its integration in both the symmetric and asymmetric cases

constraints on sensitivity characteristics in linear multivariable discrete time control systems abstract this paper is concerned with the limitations on the sensitivity characteristics for linear multivariable discrete time control systems some integral type constraints and the lower bounds of the weighted h norm imposed by the unstable poles the unstable zeros and their directions of the open loop system or the given plant are developed by a factorization approach these constraints and bounds which are tighter than those in the previous work are also characterized by the state space representations the descriptions are closely related to a special type of the algebraic riccati equation and the relation between the sampling period and the sensitivity performance is discussed for digital control systems the constraints on the more general function mixed sensitivity function are investigated in a similar way

tractability conditions for numeric csps abstract the computational complexity of the constraint satisfaction problem csp with semilinear relations over the reals has gained recent attraction as a result its complexity is known for all finite sets of semilinear relations containing the relation r x y z r 3 x y z we consider larger and more expressive classes of relations such as semialgebraic and o minimal relations we present a general result for characterising computationally hard fragments and under certain side conditions this result implies that polynomial time solvable fragments are only to be found within two limited families of sets of relations in the setting of semialgebraic relation our result takes on a simplified form and we provide a full complexity classification for constraint languages that consist of algebraic varieties full classifications like the one obtained here for algebraic varieties or the one for semilinear relations appear to be rare and we discuss several barriers for obtaining further such results these barriers have strong connections with well known open problems concerning the complexity of various restrictions of convex programming

reconstructing the topology of clones function clones are sets of functions on a fixed domain that are closed under composition and contain the projections they carry a natural algebraic structure provided by the laws of composition which hold in them as well as a natural topological structure provided by the topology of pointwise convergence under which composition of functions becomes continuous inspired by recent results indicating the importance of the topological ego of function clones even for originally algebraic problems we study questions of the following type in which situations does the algebraic structure determine the topological structure of a functional clone we pay particular attention to function clones which contain an oligomorphic permutation group and discuss applications of this situation in model theory and theoretical computer science

reasoning about temporal relations a maximal tractable subclass of allen s interval algebra we introduce a new subclass of allen s interval algebra we call ord horn subclass which is a strict superset of the pointisable subclass we prove that reasoning in the ord horn subclass is a polynomial time problem and show that the path consistency method is sufficient for deciding satisfiability further using an extensive machine generated case analysis we show that the ord horn subclass is a maximal tractable subclass of the full algebra assuming p np in fact it is the unique greatest tractable subclass amongst the subclasses that contain all basic relations

efficient search for approximate nearest neighbor in high dimensional spaces we address the problem of designing data structures that allow efficient search for approximate nearest neighbors more specifically given a database consisting of a set of vectors in some high dimensional euclidean space we want to construct a space efficient data structure that would allow us to search given a query vector for the closest or nearly closest vector in the database we also address this problem when distances are measured by the l1 norm and in the hamming cube significantly improving and extending recent results of kleinberg we construct data structures whose size is polynomial in the size of the database and search algorithms that run in time nearly linear or nearly quadratic in the dimension depending on the case the extra factors are polylogarithmic in the size of the database

rational surfaces associated with affine root systems and geometry of the painleve equations we present a geometric approach to the theory of painleve equations based on rational surfaces our starting point is a compact smooth rational surface x which has a unique anti canonical divisor d of canonical type we classify all such surfaces x to each x there corresponds a root subsystem of e 1 8 inside the picard lattice of x we realize the action of the corresponding affine weyl group as the cremona action on a family of these surfaces we show that the translation part of the affine weyl group gives rise to discrete painleve equations and that the above action constitutes their group of symmetries by backlund transformations the six painleve differential equations appear as degenerate cases of this construction in the latter context x is okamoto s space of initial conditions and d is the pole divisor of the symplectic form defining the hamiltonian structure