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36
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 86 (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...
UNLABELED (2 + 2)FREE POSETS, ASCENT SEQUENCES AND PATTERN AVOIDING PERMUTATIONS
"... Abstract. We present bijections between four classes of combinatorial objects. Two of them, the class of unlabeled (2 + 2)free posets and a certain class of chord diagrams (or involutions), already appear in the literature. The third one is a class of permutations, defined in terms of a new type of ..."
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Cited by 22 (8 self)
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Abstract. We present bijections between four classes of combinatorial objects. Two of them, the class of unlabeled (2 + 2)free posets and a certain class of chord diagrams (or involutions), already appear in the literature. The third one is a class of permutations, defined in terms of a new type of pattern. An attractive property of these patterns is that, like classical patterns, they are closed under the action of D8, the symmetry group of the square. The fourth class is formed by certain integer sequences, called ascent sequences, which have a simple recursive structure and are shown to encode (2 + 2)free posets, chord diagrams and permutations. Our bijections preserve numerous statistics. We also determine the generating function of these classes of objects, thus recovering a series obtained by Zagier for chord diagrams. That this series also counts (2 + 2)free posets seems to be new. Finally, we characterize the ascent sequences that correspond to permutations avoiding the barred pattern 3¯152¯4, and enumerate those permutations, thus settling a conjecture of Lara Pudwell. 1.
Temporal Constraints: A Survey
, 1998
"... . Temporal Constraint Satisfaction is an information technology useful for representing and answering queries about the times of events and the temporal relations between them. Information is represented as a Constraint Satisfaction Problem (CSP) where variables denote event times and constraints re ..."
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Cited by 20 (1 self)
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. Temporal Constraint Satisfaction is an information technology useful for representing and answering queries about the times of events and the temporal relations between them. Information is represented as a Constraint Satisfaction Problem (CSP) where variables denote event times and constraints represent the possible temporal relations between them. The main tasks are two: (i) deciding consistency, and (ii) answering queries about scenarios that satisfy all constraints. This paper overviews results on several classes of Temporal CSPs: qualitative interval, qualitative point, metric point, and some of their combinations. Research has progressed along three lines: (i) identifying tractable subclasses, (ii) developing exact search algorithms, and (iii) developing polynomialtime approximation algorithms. Most available techniques are based on two principles: (i) enforcing local consistency (e.g. pathconsistency), and (ii) enhancing naive backtracking search. Keywords: Temporal Constra...
Graphs and partially ordered sets: recent results and new directions
 JACOBSON (EDS.), SURVEYS IN GRAPH THEORY, CONGRESSUS NUMERANTIUM
, 1996
"... We survey some recent research progress on topics linking graphs and finite partially ordered sets. Among these topics are planar graphs, hamiltonian cycles and paths, graph and hypergraph coloring, online algorithms, intersection graphs, inclusion orders, random methods and ramsey theory. In each ..."
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Cited by 9 (2 self)
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We survey some recent research progress on topics linking graphs and finite partially ordered sets. Among these topics are planar graphs, hamiltonian cycles and paths, graph and hypergraph coloring, online algorithms, intersection graphs, inclusion orders, random methods and ramsey theory. In each case, we discuss open problems and future research directions.
Ramsey theory and sequences of random variables
 COMBINATORICS, PROBABILITY AND COMPUTING
, 1998
"... We consider probability spaces which contain a family {EA: A ⊆{1,2,...,n}, A  = k} of events indexed by the kelement subsets of {1, 2,...,n}. A pair (A, B) of kelement subsets of {1, 2,...,n} is called a shift pair if the largest k − 1 elements of A coincide with the smallest k − 1 elements of ..."
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Cited by 8 (3 self)
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We consider probability spaces which contain a family {EA: A ⊆{1,2,...,n}, A  = k} of events indexed by the kelement subsets of {1, 2,...,n}. A pair (A, B) of kelement subsets of {1, 2,...,n} is called a shift pair if the largest k − 1 elements of A coincide with the smallest k − 1 elements of B. For a shift pair (A, B), Pr[AB] is the probability that event EA is true and EB is false. We investigate how large the minimum value of Pr[AB], taken over all shift pairs, can be. As n →∞, this value converges to a number λk, with
The ongoing dialog between empirical science and measurement theory
 Journal of Mathematical Psychology
, 1996
"... This review article attempts to highlight from my personal perspective some of the major developments in the representational theory of measurement during the past 50 years. Emphasis is placed on the ongoing interplay between the development of abstract theory and the attempts to apply it to empiric ..."
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Cited by 8 (0 self)
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This review article attempts to highlight from my personal perspective some of the major developments in the representational theory of measurement during the past 50 years. Emphasis is placed on the ongoing interplay between the development of abstract theory and the attempts to apply it to empirically testable phenomena. The article has four major sections. The first concerns classical representational measurement, which was the successful attempt to formulate the major measurement methods of classical physics: extensive and additive conjoint structures, their distributive interlock in dimensional analysis, and intensive (averaging) structures. The second illustrates a nontrivial behavioral example using both extensive and conjoint measurement plus functional equations to arrive at rank and signdependent utility (also called cumulative prospect) representations for decision making under risk. The third section, contemporary representational measurement, somewhat overlaps the classical one but includes new findings and approaches: representations of nonadditive concatenation and conjoint structures; a general theory of scale types; results for general, finitely unique, homogeneous structures; structures that are homogeneous between singular points; generalized distributive triples; and a generalization of dimensional analysis to include any ratio scalable attribute; and the concept of meaningfulness. The final section concerns applications of the latter ideas to psychophysical scaling and merging functions.] 1996 Academic Press, Inc. 1.
New perspectives on interval orders and interval graphs
 in Surveys in Combinatorics
, 1997
"... Abstract. Interval orders and interval graphs are particularly natural examples of two widely studied classes of discrete structures: partially ordered sets and undirected graphs. So it is not surprising that researchers in such diverse fields as mathematics, computer science, engineering and the so ..."
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Cited by 7 (5 self)
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Abstract. Interval orders and interval graphs are particularly natural examples of two widely studied classes of discrete structures: partially ordered sets and undirected graphs. So it is not surprising that researchers in such diverse fields as mathematics, computer science, engineering and the social sciences have investigated structural, algorithmic, enumerative, combinatorial, extremal and even experimental problems associated with them. In this article, we survey recent work on interval orders and interval graphs, including research on online coloring, dimension estimates, fractional parameters, balancing pairs, hamiltonian paths, ramsey theory, extremal problems and tolerance orders. We provide an outline of the arguments for many of these results, especially those which seem to have a wide range of potential applications. Also, we provide short proofs of some of the more classical results on interval orders and interval graphs. Our goal is to provide fresh insights into the current status of research in this area while suggesting new perspectives and directions for the future. 1.
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...