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12
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...
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 contains ex ..."
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Cited by 29 (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 performance evaluation of CP list scheduling heuristics for communication intensive task graphs
 In IPPS/SPDP’98
, 1998
"... Listbased priority schedulers have long been one of the dominant classes of static scheduling algorithms. Such heuristics have been predominantly based around the “critical path, most immediate successors first” (CP/MISF) priority. The ability of this type of scheduler to handle increased levels of ..."
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Cited by 11 (0 self)
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Listbased priority schedulers have long been one of the dominant classes of static scheduling algorithms. Such heuristics have been predominantly based around the “critical path, most immediate successors first” (CP/MISF) priority. The ability of this type of scheduler to handle increased levels of communication overhead is examined in this paper. Three of the more popular list scheduling heuristics, HLFET [1] and ISH and DSH [10], plus the Mapping Heuristic [4,6] are subjected to a performance based comparison, with results demonstrating their inadequacies in communicationintensive cases. Performance degradation in these instances is partly due to the level alteration problem, but more significantly to conservative estimation of communication costs due to the assumption of zero link contention. The significance of this component of communication is also examined in this paper. 1.
Recognition Algorithms for Orders of Small Width and Graphs of Small Dilworth Number
"... Partially ordered sets of small width and graphs of small Dilworth number have many interesting properties and have been well studied. Here we show that recognition of such orders and graphs can be done more eÆciently than by using the wellknown algorithms based on bipartite matching and matrix mul ..."
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Cited by 9 (0 self)
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Partially ordered sets of small width and graphs of small Dilworth number have many interesting properties and have been well studied. Here we show that recognition of such orders and graphs can be done more eÆciently than by using the wellknown algorithms based on bipartite matching and matrix multiplication. In particular, we show that deciding deciding if an order has width k can be done in O(kn²) time and whether a graph has Dilworth number k can be done in O(k²n²) time. For very small k we have even better results. We show that orders of width at most 3 can be recognized in O(n) time and of width at most 4 in O(n log n).
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 2 (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 Performance Evaluation of CP List Scheduling Heuristics for Communication Intensive Task Graphs
, 1998
"... Listbased priority schedulers have long been one of the dominant classes of static scheduling algorithms. Such heuristics have been predominantly based around the "critical path, most immediate successors first" (CP/MISF) priority. The ability of this type of scheduler to handle increased levels of ..."
Abstract
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Listbased priority schedulers have long been one of the dominant classes of static scheduling algorithms. Such heuristics have been predominantly based around the "critical path, most immediate successors first" (CP/MISF) priority. The ability of this type of scheduler to handle increased levels of communication overhead is examined in this paper. Three of the more popular list scheduling heuristics, HLFET [1] and ISH and DSH [10], plus the Mapping Heuristic [4,6] are subjected to a performance based comparison, with results demonstrating their inadequacies in communicationintensive cases. Performance degradation in these instances is partly due to the level alteration problem, but more significantly to conservative estimation of communication costs due to the assumption of zero link contention. The significance of this component of communication is also examined in this paper. 1. Introduction Task scheduling is one of the most challenging problems facing parallel programmers today....
Cuts and Connectivity in Chordal Graphs
"... A cut (A; B) in a graph G is called internal, i N(A) 6= B and N(B) 6= A. In this paper, we study the structure of internal cuts in chordal graphs. We show that if (A; B) is an internal cut in a chordal graph, then for each i, 0 i (G)+1, there exists a clique K i such that jK i j = (G)+1, jK i ..."
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A cut (A; B) in a graph G is called internal, i N(A) 6= B and N(B) 6= A. In this paper, we study the structure of internal cuts in chordal graphs. We show that if (A; B) is an internal cut in a chordal graph, then for each i, 0 i (G)+1, there exists a clique K i such that jK i j = (G)+1, jK i T Aj = i and jK i T Bj = k+1 i, where (G) is the vertex connectivity of G. In general, there can be an exponential number of internal cuts in a chordal graph, while the number of maximal cliques can be at most n (G) 1 [2]. Also there exists chordal graphs, all of whose maximal cliques are of size (G) + 1. Thus, above result throws some light as to the way the cliques are arranged in chordal graphs, with respect to their cuts. We also show that in a chordal graph G, every internal cut should contain at least (G)((G)+1) 2 edges. This lower bound is tight, in the sense that there exists chordal graphs with internal cuts having exactly (G)((G)+1) 2 edges.