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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...
Optimal FPGA Module Placement with Temporal Precedence Constraints
 IN PROC. DATE 2001, DESIGN, AUTOMATION AND TEST IN EUROPE
, 2001
"... We consider the optimal placement of hardware modules in space and time for FPGA architectures with reconfiguration capabilities, where modules are modeled as threedimensional boxes in space and time. Using a graphtheoretic characterization of feasible packings, we are able to solve the following p ..."
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Cited by 38 (5 self)
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We consider the optimal placement of hardware modules in space and time for FPGA architectures with reconfiguration capabilities, where modules are modeled as threedimensional boxes in space and time. Using a graphtheoretic characterization of feasible packings, we are able to solve the following problems: (a) Find the minimal execution time of the given problem on an FPGA of fixed size, (b) Find the FPGA of minimal size to accomplish the tasks within a fixed time limit. Furthermore, our approach is perfectly suited for the treatment of precedence constraints for the sequence of tasks, which are present in virtually all practical instances. Additional mathematical structures are developed that lead to a powerful framework for computing optimal solutions. The usefulness is illustrated by computational results.
Multidimensional packing with order constraints
 in Proceedings 7th International Workshop on Algorithms and Data Structures, vol. 2125 HIGHERDIMENSIONAL PACKING WITH ORDER CONSTRAINTS 21
"... Abstract. We present a first exact study on higherdimensional packing problems with order constraints. Problems of this type occur naturally in applications such as logistics or computer architecture and can be interpreted as higherdimensional generalizations of scheduling problems. Using graphth ..."
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Cited by 8 (3 self)
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Abstract. We present a first exact study on higherdimensional packing problems with order constraints. Problems of this type occur naturally in applications such as logistics or computer architecture and can be interpreted as higherdimensional generalizations of scheduling problems. Using graphtheoretic structures to describe feasible solutions, we develop a novel exact branchandbound algorithm. This extends previous work by Fekete and Schepers; a key tool is a new ordertheoretic characterization of feasible extensions of a partial order to a given complementarity graph that is tailormade for use in a branchandbound environment. The usefulness of our approach is validated by computational results. Key words. higherdimensional packing, higherdimensional scheduling, reconfigurable computing, precedence constraints, exact algorithms, modular decomposition.
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...
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...
Extending Partial Suborders
, 2001
"... We consider the problem of finding a transitive orientation T of a comparability graph G = (V, E), such that a given partial order P is extended. Existing algorithms for this problem require the full knowledge of E, so they are of limited use in the context of a branchandbound algorithm, where onl ..."
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Cited by 1 (1 self)
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We consider the problem of finding a transitive orientation T of a comparability graph G = (V, E), such that a given partial order P is extended. Existing algorithms for this problem require the full knowledge of E, so they are of limited use in the context of a branchandbound algorithm, where only parts of E may be known at any stage. We present a new approach to the problem by describing a pair of necessary and sufficient conditions for the existence of an orientation T , based on two simple forbidden subconfigurations. This allows it to solve higherdimensional packing and scheduling problems of interesting size to optimality. We have implemented this approach and the computational results are convincing.
Path Graphs and PRtrees
, 2012
"... The PRtree data structure is introduced to characterize the sets of pathtree models of path graphs. We further characterize the sets of directed pathtree models of directed path graphs with a slightly restricted form of the PRtree called the Strong PRtree. Additionally, via PRtrees and Strong ..."
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Cited by 1 (1 self)
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The PRtree data structure is introduced to characterize the sets of pathtree models of path graphs. We further characterize the sets of directed pathtree models of directed path graphs with a slightly restricted form of the PRtree called the Strong PRtree. Additionally, via PRtrees and Strong PRtrees, we characterize path graphs and directed path graphs by their Split Decompositions. Two distinct approaches (Split Decomposition and Reduction) are presented to construct a PRtree that captures the pathtree models of a given graph G = (V,E) with n = V and m = E. An implementation of the split decomposition approach is presented which runs in O(nm) time. Similarly, an implementation of the reduction approach is presented which runs in O(A(n + m)nm) time (where A(s) is the inverse of Ackermann’s function arising from UnionFind [40]). Also, from a PRtree, an algorithm to construct a corresponding Strong PRtree is given which runs in O(n +m) time. The sizes of the PRtrees and Strong PRtrees produced by these approaches are O(n + m) with respect to the given graph. Furthermore, we demonstrate that an implicit form of the PRtree and Strong PRtree can be represented in O(n) space.
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...