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14
Complexity and Algorithms for Reasoning About Time: A GraphTheoretic Approach
, 1992
"... 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 ..."
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Cited by 100 (11 self)
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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 (interval) 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 NPcomplete. On the positive side, we give efficient algorithms for several restrictions of the problem. In the process, the interval graph sandwich problem is introduced, and is shown to be NPcomplete. This problem is als...
Reasoning About Temporal Relations: The Tractable Subalgebras Of Allen's Interval Algebra
 Journal of the ACM
, 2001
"... Allen's interval algebra is one of the best established formalisms for temporal reasoning. This paper is the final step in the classification of complexity in Allen's algebra. We show that the current knowledge about tractability in the interval algebra is complete, that is, this algebra c ..."
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Cited by 42 (2 self)
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Allen's interval algebra is one of the best established formalisms for temporal reasoning. This paper is the final step in the classification of complexity in Allen's algebra. We show that the current knowledge about tractability in the interval algebra is complete, that is, this algebra contains exactly eighteen maximal tractable subalgebras, and reasoning in any fragment not entirely contained in one of these subalgebras is NPcomplete. We obtain this result by giving a new uniform description of the known maximal tractable subalgebras and then systematically using an algebraic technique for identifying maximal subalgebras with a given property.
A hyperbolic cosine latent trait model for unfolding dichotomous singlestimulus responses
 Applied Psychological Measurement
, 1993
"... The hyperbolic cosine unfolding model for direct responses of persons to individual stimuli is elaborated in three ways. First, the parameter of the stimulus, which reflects a region within which people located there are more likely to respond positively than negatively, is shown to be a property ..."
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Cited by 24 (0 self)
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The hyperbolic cosine unfolding model for direct responses of persons to individual stimuli is elaborated in three ways. First, the parameter of the stimulus, which reflects a region within which people located there are more likely to respond positively than negatively, is shown to be a property of the data and not arbitrary as first supposed. Second, the model is used to construct a related model for pairwise preferences. This model, for which joint maximum likelihood estimates are derived, satisfies strong stochastic transitivity. Third, the role of substantive theory in evaluating the fit between the data and the models, in which unique solutions for the estimates are not guaranteed, is explored by analyzing responses of one group of persons to a single set of stimuli
Preference modelling
 State of the Art in Multiple Criteria Decision Analysis
, 2005
"... This paper provides the reader with a presentation of preference modelling fundamental notions as well as some recent results in this field. Preference modelling is an inevitable step in a variety of fields: economy, sociology, psychology, mathematical programming, even medicine, archaeology, and ob ..."
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Cited by 18 (1 self)
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This paper provides the reader with a presentation of preference modelling fundamental notions as well as some recent results in this field. Preference modelling is an inevitable step in a variety of fields: economy, sociology, psychology, mathematical programming, even medicine, archaeology, and obviously decision analysis. Our notation and some basic definitions, such as those of binary relation, properties and ordered sets, are presented at the beginning of the paper. We start by discussing different reasons for constructing a model or preference. We then go through a number of issues that influence the construction of preference models. Different formalisations besides classical logic such as fuzzy sets and nonclassical logics become necessary. We then present different types of preference structures reflecting the behavior of a decisionmaker: classical, extended and valued ones. It is relevant to have a numerical representation of preferences: functional representations, value functions. The concepts of thresholds and minimal representation are also introduced in this section. In section 7, we briefly explore the concept of deontic logic (logic of preference) and other formalisms associated with &quot;compact representation of preferences &quot; introduced for special purposes. We end the paper with some concluding remarks.
Satisfiability Problems on Intervals and Unit Intervals
 Theoretical Computer Science
, 1997
"... For an interval graph with some additional order constraints between pairs of nonintersecting 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 constrain ..."
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Cited by 5 (1 self)
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For an interval graph with some additional order constraints between pairs of nonintersecting 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 NPcompleteness of another restriction. 1 Introduction Two intervals x; y on the real line may either intersect or one of them is completely to the left of the othe...
A generic disjunctive/conjunctive decomposition model for naxy relations
 Journal of Mathematical Psychology
, 1999
"... This paper discusses a generic decomposition model that represents an arbitrary nary relation as a disjunctive or conjunctive combination of a number of nary component relations of a prespecified type. An important subclass of orderpreserving decompositions is defined and its properties are deriv ..."
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Cited by 4 (3 self)
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This paper discusses a generic decomposition model that represents an arbitrary nary relation as a disjunctive or conjunctive combination of a number of nary component relations of a prespecified type. An important subclass of orderpreserving decompositions is defined and its properties are derived. The generic model is shown to subsume various known models as special cases, including the models of Boolean factor analysis, hierarchical classes analysis, and disjunctiveconjunctive nonmetric factor analysis. Moreover, it also subsumes a broad range of new models as exemplified with a novel model for multidimensional parallelogram analysis and novel threeway extensions of nonmetric factor analysis. 1999 Academic Press
Interval Graphs with Side (and Size) Constraints
 In Proc. of the Third Annual European Symp. on Algorithms, (ESA 95) Corfu, Greece
, 1995
"... . We study problems of determining whether a given interval graph has a realization which satisfies additional given constraints. Such problems occur frequently in applications where entities are modeled as intervals along a line (events along a time line, DNA segments along a chromosome, etc.). ..."
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Cited by 3 (1 self)
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. We study problems of determining whether a given interval graph has a realization which satisfies additional given constraints. Such problems occur frequently in applications where entities are modeled as intervals along a line (events along a time line, DNA segments along a chromosome, etc.). When the additional information is order constraints on pairs of disjoint intervals, we give a linear time algorithm. Extant algorithms for this problem (known also as seriation with side constraints) required quadratic time. When the constraints are bounds on distances between endpoints, and the graph admits a unique clique order, we show that the problem is polynomial. However, we show that even when the lengths of all intervals are precisely predetermined, the problem is NPcomplete. We also study unit interval satisfiability problems, which are concerned with the realizability of a set of unit intervals along a line, subject to precedence and intersection constraints. For all po...
Realizing Interval Graphs With Size And Distance Constraints
 SIAM Journal on Discrete Mathematics
, 1997
"... . We study the following problem: Given an interval graph, does it have a realization which satisfies additional constraints on the distances between interval endpoints? This problem arises in numerous applications in which topological information on intersection of pairs of intervals is accompanied ..."
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Cited by 3 (0 self)
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. We study the following problem: Given an interval graph, does it have a realization which satisfies additional constraints on the distances between interval endpoints? This problem arises in numerous applications in which topological information on intersection of pairs of intervals is accompanied by additional metric information on their order, distance or sizes. An important application is physical mapping, a central challenge in the human genome project. Our results are: (1) A polynomial algorithm for the problem on interval graphs which admit a unique clique order (UCO graphs). This class of graphs properly contains all prime interval graphs. (2) In case all constraints are upper and lower bounds on individual interval lengths, the problem on UCO graphs is linearly equivalent to deciding if a system of difference inequalities is feasible. (3) Even if all the constraints are prescribed lengths of individual intervals, the problem is NPcomplete. Hence, problems (1) and (2) are als...
A Conjunctive Parallelogram Model for Pick Any N Data.” Psychometrika 69:401–420
, 2004
"... This paper proposes a multidimensional generalization f Coombs ' (1964) parallelogram odel for "pick any/n " data, which result from each of a number of subjects having selected a number of objects (s)he likes most from a prespecified set of n objects. In the model, persons and object ..."
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This paper proposes a multidimensional generalization f Coombs ' (1964) parallelogram odel for "pick any/n " data, which result from each of a number of subjects having selected a number of objects (s)he likes most from a prespecified set of n objects. In the model, persons and objects are represented in a low dimensional space defined by a set of ordinal variables with a prespecified number of categories; objects axe represented as points and persons as intervals on each dimension. A conjunctive combination rule is assumed implying that a person selects an object if and only if the object is within the subject's interval on each dimension. An algorithm for fitting the model to a data set is presented and evaluated in a simulation study. The model is illustrated with data on preferences regarding holiday trips. Key words: Choice data analysis, binary data, parallelogram analysis, noncompensatory combination rule. A natural and widely used method to obtain information on an individual's preferences with respect o a set of n choice objects involves presenting him with the full set of objects and having him select he subset of objects he likes most. When this method is used to have each of h persons elect from the same set of n objects, it gives rise to "pick any/n " data (Coombs, 1964), which can be represented by an h x n binary matrix D, with dij = 1 if person i selects object j and 0 otherwise. This paper presents a new model for such data, which can be considered a
Interval Graphs with Side Constraints
, 1995
"... We study problems of determining whether a given interval graph has a realization which satisfies additional given constraints. Such problems occur frequently in applications where entities are modeled as intervals along a line (events along a time line, DNA segments along a chromosome, etc.). When ..."
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We study problems of determining whether a given interval graph has a realization which satisfies additional given constraints. Such problems occur frequently in applications where entities are modeled as intervals along a line (events along a time line, DNA segments along a chromosome, etc.). When the additional information is order constraints on pairs of disjoint intervals, we give a linear time algorithm. Extant 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. When the constraints are bounds on distances between endpoints, and the graph admits a unique clique order, we show that the problem is polynomial. The special case of this problem where the constraints are bounds on interval length is shown to be linearly equivalent to deciding if a system of difference inequalities is feasible. However, we show that even when the lengths of all intervals ar...