duality and nonlinear graph laplacians abstract in this paper we show that a recent nearly linear time algorithm for solving a system of equations arising from a graph laplacian can be extended to a large class of nonlinear systems of equations based on a nonlinear generalization of the graph laplacian this result follows from a nonlinear generalization of the notable duality between graph laplacians and electrical flows beyond providing a fast algorithm for a class of nonlinear sets of equations our work highlights the robustness of spectral graph theory and suggests new directions for research in spectral graph theory specifically we present an iterative algorithm for solving a class of nonlinear system of equations arising from a nonlinear generalization of the graph laplacian in o k 2 m log k n iterations where k is a measure of nonlinearity n is the number of variables m is the number of nonzero entries in the generalized graph laplacian l is the solution accuracy and o neglects non leading logarithmic terms this algorithm is a natural nonlinear extension of the one in kelner et al 2013 which solves a linear laplacian system of equations in nearly linear time unlike the linear case in the nonlinear case each iteration takes o n time so the total running time is o k 2 m n log k n for sparse graphs with m o n and fixed k this nonlinear algorithm is o n 2 log n which is slightly faster than standard methods for solving linear equations which require approximately o n 2 38 time

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

incremental qualitative temporal reasoning algorithms for the point algebra and the ord horn class in many applications of temporal reasoning we are interested in processing temporal information incrementally in particular given a set of temporal constraints a temporal csp and a new constraint we want to maintain certain properties of the extended temporal csp e g a solution rather than recomputing them from scratch the point algebra pa and the interval algebra ia are two well known frameworks for qualitative temporal reasoning the reasoning algorithms for pa and the tractable fragments of ia such as nebel and burckert s maximal tractable class of relations ord horn have originally been designed for static reasoning in this paper we study the incremental version of the fundamental reasoning problems in the context of these tractable classes we propose a collection of new polynomial algorithms that can amortize their complexity when processing a sequence of input constraints to incrementally decide satisfiability to maintain a solution or to update the minimal representation of the csp our incremental algorithms improve the total time complexity of using existing static techniques by a factor of o n or o n2 where n is the number of the variables involved by the temporal csp an experimental analysis focused on constraints over pa confirms the computational advantage of our incremental approach

computational complexity of relating time points with interval abstract several algebras have been proposed for reasoning about qualitative constraints over the time line one of these algebras is vilain s point interval algebra which can relate time points with time intervals apart from being a stand alone qualitative algebra it is also used as a subalgebra in meiri s approach to temporal reasoning which combines reasoning about metric and qualitative temporal constraints over both time points and time intervals while the satisfiability problem for the full point interval algebra is known to be np complete not much is known about its 4 294 967 296 subclasses this article completely determines the computational complexity of these subclasses and it identifies all of the maximal tractable subalgebras five in total

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

a complete classification of tractability in allen s algebra relative to subsets of basic relations abstract we characterise the set of subalgebras of allen s algebra which have a tractable satisfiability problem and in addition contain certain basic relations the conclusion is that no tractable subalgebra dial is not known in the literature can contain more than the three basic relations b and b where b d o s f this means that concerning algebras for specifying complete knowledge about temporal information there is no hope of finding yet unknown classes with much expressivity we also classify completely some cases where we cannot even express complete information but close to complete showing that there are exactly two maximal tractable algebras containing the relation exactly two containing the relation m m and exactly three containing the relation m the algebras containing can express the notion of sequentiality thus we have a complete characterization of tractable inference using that notion

querying temporal and spatial constraint networks in ptime we start with the assumption that temporal and spatial knowledge usually captured by constraint networks can be represented and queried more effectively by using the scheme of indefinite constraint databases because query evaluation in this scheme is in general a hard computational problem we seek tractable instances of query evaluation we assume that we have a class of constraints c with some reasonable computational and closure properties the computational properties of interest are that the satisfiability problem and an appropriate version of the variable elimination problem for c should be solvable in ptime under this assumption we exhibit general classes of indefinite constraint databases and first order modal queries for which query evaluation can be done with ptime data complexity we then search for tractable instances of c among the subclasses of horn disjunctive linear constraints over the rationals from previous research we know that the satisfiability problem for horn disjunctive linear constraints is solvable in ptime but not the variable elimination problem thus we try to discover subclasses of horn disjunctive linear constraints with tractable variable elimination problems the class of utvpi constraints is the largest class that we show to have this property finally we restate our general tractability results with c ranging over the newly discovered tractable classes interesting tractable query answering problems for indefinite temporal and spatial constraint databases are identified in this way we close our complexity analysis by precisely outlining the frontier between tractable and possibly intractable query answering problems

reasoning about cardinal directions between extended objects the np hardness result the cardinal direction calculus cdc proposed by goyal and egenhofer is a very expressive qualitative calculus for directional information of extended objects early work has shown that consistency checking of complete networks of basic cdc constraints is tractable while reasoning with the cdc in general is np hard this paper shows however that if some constraints are unspecified then consistency checking of incomplete networks of basic cdc constraints is already intractable this draws a sharp boundary between the tractable and intractable subclasses of the cdc the result is achieved by a reduction from the well known 3 sat problem

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

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

the maximal subgroups of the baby monster i abstract in this paper we describe the completion of the determination of the maximal subgroups of the baby monster simple group our results are proved using computer calculations with matrix generators for the group the full details of the calculations will appear elsewhere

relation algebras of intervals abstract given a representation of a relation algebra we construct relation algebras of pairs and of intervals if the representation happens to be complete homogeneous and fully universal then the pair and interval algebras can be constructed direct from the relation algebra if further the original relation algebra is categorical we show that the interval algebra is too the complexity of relation algebras is studied and it is shown that every pair algebra with infinite representations is intractable applications include constructing an interval algebra that combines metric and interval expressivity

spatial reasoning with rcc 8 and connectedness constraints in euclidean spaces the language rcc 8 is a widely studied formalism for describing topological arrangements of spatial regions the variables of this language range over the collection of non empty regular closed sets of n dimensional euclidean space here denoted rc r n and its non logical primitives allow us to specify how the interiors exteriors and boundaries of these sets intersect the key question is the satisfiability problem given a finite set of atomic rcc 8 constraints in m variables determine whether there exists an m tuple of elements of rc r n satisfying them these problems are known to coincide for all n 1 so that rcc 8 satisfiability is independent of dimension this common satisfiability problem is nlogspace complete unfortunately rcc 8 lacks the means to say that a spatial region comprises a single piece and the present article investigates what happens when this facility is added we consider two extensions of rcc 8 rcc 8 c in which we can state that a region is connected and rcc 8 c in which we can instead state that a region has a connected interior the satisfiability problems for both these languages are easily seen to depend on the dimension n for n 3 furthermore in the case of rcc 8 c we show that there exist finite sets of constraints that are satisfiable over rc r 2 but only by wild regions having no possible physical meaning this prompts us to consider interpretations over the more restrictive domain of non empty regular closed polyhedral sets rcp r n we show that a the satisfiability problems for rcc 8 c equivalently rcc 8 c over rc r and rcp r are distinct and both np complete b the satisfiability problems for rcc 8 c over rc r 2 and rcp r 2 are identical and np complete c the satisfiability problems for rcc 8 c over rc r 2 and rcp r 2 are distinct and the latter is np complete decidability of the satisfiability problem for rcc 8 c over rc r 2 is open for n 3 rcc 8 c and rcc 8 c are not interestingly different from rcc 8 we finish by answering the following question given that a set of rcc 8 c or rcc 8 c constraints is satisfiable over rc r n or rcp r n how complex is the simplest satisfying assignment in particular we exhibit for both languages a sequence of constraints n satisfiable over rcp r 2 such that the size of n grows polynomially in n while the smallest configuration of polygons satisfying n cuts the plane into a number of pieces that grows exponentially we further show that over rc r 2 rcc 8 c again requires exponentially large satisfying diagrams while rcc 8 c can force regions in satisfying configurations to have infinitely many components

combining qualitative and quantitative constraints in temporal reasoning abstract this paper presents a general model for temporal reasoning that is capable of handling both qualitative and quantitative information this model allows the representation and processing of many types of constraints discussed in the literature to date including metric constraints restricting the distance between time points and qualitative disjunctive constraints specifying the relative position of temporal objects reasoning tasks in this unified framework are formulated as constraint satisfaction problems and are solved by traditional constraint satisfaction techniques such as backtracking and path consistency new classes of tractable problems are characterized involving qualitative networks augmented by quantitative domain constraints some of which can be solved in polynomial time using arc and path consistency

complexity classification in qualitative temporal constraint reasoning we study the computational complexity of the qualitative algebra which is a temporal constraint formalism that combines the point algebra the point interval algebra and allen s interval algebra we identify all tractable fragments and show that every other fragment is np complete

complexity and algorithms for reasoning about time a graph theoretic approach temporal events are regarded here as intervals on a time line this paper deals with problems in reasoning about such intervals when the precise topological relationship between them is unknown or only partially specified this work unifies notions of interval algebras in artificial intelligence with those of interval orders and interval graphs in combinatorics the satisfiability minimal labeling all solutions and all realizations problems are considered for temporal internal data several versions are investigated by restricting the possible interval relationships yielding different complexity results we show that even when the temporal data comprises of subsets of relations based on intersection and precedence only the satisfiability question is np complete

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

satisfiability problems on intervals and unit intervals abstract for an interval graph with some additional order constraints between pairs of non intersecting intervals we give a linear time algorithm to determine if there exists a realization which respects the order constraints previous algorithms for this problem known also as seriation with side constraints required quadratic time this problem contains as subproblems interval graph and interval order recognition on the other hand it is a special case of the interval satisfiability problem which is concerned with the realizability of a set of intervals along a line subject to precedence and intersection constraints we study such problems for all possible restrictions on the types of constraints when all intervals must have the same length we give efficient algorithms for several restrictions of the problem and show the np completeness of another restriction

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

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

tractable disjunctions of linear constraints basic results and applications to temporal reasoning we study the problems of deciding consistency and performing variable elimination for disjunctions of linear inequalities and disequations with at most one inequality per disjunction this new class of constraints extends the class of generalized linear constraints originally studied by lassez and mcaloon we show that deciding consistency of a set of constraints in this class can be done in polynomial time we also present a variable elimination algorithm which is similar to fourier s algorithm for linear inequalities finally we use these results to provide new temporal reasoning algorithms for the ord horn subclass of allen s interval formalism we also show that there is no low level of local consistency that can guarantee global consistency for the ord horn subclass this property distinguishes the ord horn subclass from the pointizable subclass for which strong 5 consistency is sufficient to guarantee global consistency and the continuous endpoint subclass for which strong 3 consistency is sufficient to guarantee global consistency copyright 2001 elsevier science b v

nonsingular curves passing through points of p2 in generic position i abstract the least integer is looked for such that there exists a nonsingular curve of degree less than or equal to passing through any assigned set of n points of p 2 in generic position here generic takes different meanings and the generic n tuples of points describe nonempty open subsets u of p 2 n indeed we find i for u 1 n tuples in general position u 1 n 2 ii if n d 2 2 h 0 h d 1 for u2 n tuples in n generic position iii with the notations of ii for u3 n tuples in uniform position in any case a nonempty open subset u u3 is found in which u d 1

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

temporal reasoning about fuzzy intervals traditional approaches to temporal reasoning assume that time periods and time spans of events can be accurately represented as intervals real world time periods and events on the other hand are often characterized by vague temporal boundaries requiring appropriate generalizations of existing formalisms this paper presents a framework for reasoning about qualitative and metric temporal relations between vague time periods in particular we show how several interesting problems like consistency and entailment checking can be reduced to reasoning tasks in existing temporal reasoning frameworks we furthermore demonstrate that all reasoning tasks of interest are np complete which reveals that adding vagueness to temporal reasoning does not increase its computational complexity to support efficient reasoning a large tractable subfragment is identified among others generalizing the well known ord horn subfragment of the interval algebra extended with metric constraints

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

a unifying approach to temporal constraint reasoning abstract we present a formalism disjunctive linear relations dlrs for reasoning about temporal constraints dlrs subsume most of the formalisms for temporal constraint reasoning proposed in the literature and is therefore computationally expensive we also present a restricted type of dlrs horn dlrs which have a polynomial time satisfiability problem we prove that most approaches to tractable temporal constraint reasoning can be encoded as horn dlrs including the ord horn algebra by nebel and burckert and the simple temporal constraints by dechter et al thus dlrs is a suitable unifying formalism for reasoning about temporal constraints

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

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

the complexity of query evaluation in indefinite temporal constraint databases abstract in previous work we have developed the scheme of indefinite l constraint databases where l the parameter is a first order constraint language this scheme extends the constraint database proposal of kanellakis et al 1990 1995 to include indefinite or uncertain information in the style of imielinski and lipski 1984 in this paper we study the complexity of query evaluation in an important instance of this abstract scheme indefinite temporal constraint databases our results indicate that the data combined complexity of query evaluation does not change when we move from queries in relational calculus over relational databases to queries in relational calculus with temporal constraints over temporal constraint databases this fact remains true even when we consider query evaluation in relational databases with indefinite information vs query evaluation in indefinite temporal constraint databases in the course of our work we provide precise bounds on the complexity of decision quantifier elimination for a subtheory of presburger arithmetic and a subtheory of real addition with order the bounds for the latter theory are original and of independent interest

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

backtracking algorithms for disjunctions of temporal constraints abstract we extend the framework of simple temporal problems studied originally by dechter meiri and pearl to consider constraints of the form x 1 y 1 r 1 x n y n r n where x 1 x n y 1 y n are variables ranging over the real numbers r 1 r n are real constants and n 1 this is a wide class of temporal constraints that can be used to model a variety of problems in temporal reasoning scheduling planning and temporal constraint databases we have implemented several progressively more efficient algorithms for the consistency checking problem for this class of temporal constraints backtracking backjumping three variations of forward checking and forward checking with backjumping we have partially ordered the above algorithms according to the number of visited search nodes and the number of performed consistency checks although our problem is non binary our results agree with the results of kondrak and van beek who consider only binary constraints we have also studied the performance of the above algorithms experimentally using randomly generated sets of data and job shop scheduling problems the experiments with random instances allowed us to locate the hard region for this class of problems the results show that hard problems occur at a critical value of the ratio of disjunctions to variables

determining the consistency of partial tree descriptions we present an efficient algorithm that decides the consistency of partial descriptions of ordered trees the constraint language of these descriptions was introduced by cornell in computational linguistics the constraints specify for pairs of nodes sets of admissible relative positions in an ordered tree cornell asked for an algorithm to find a tree structure satisfying these constraints this computational problem generalizes the common supertree problem studied in phylogenetic analysis and also generalizes the network consistency problem of the so called left linear point algebra we present the first polynomial time algorithm for cornell s problem which runs in time o mn where m is the number of constraints and n the number of variables in the constraint

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