# Complexity and algorithms for reasoning about time: a graph-theoretic approach

@article{Golumbic1993ComplexityAA, title={Complexity and algorithms for reasoning about time: a graph-theoretic approach}, author={M. Golumbic and R. Shamir}, journal={J. ACM}, year={1993}, volume={40}, pages={1108-1133} }

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… Expand

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#### References

SHOWING 1-10 OF 82 REFERENCES

Reasoning About Qualitative Temporal Information

- Computer Science, Mathematics
- Artif. Intell.
- 1992

This paper defines computationally efficient procedures for solving two related reasoning tasks that arise in interval and point algebras: Given (possibly indefinite) knowledge of the relationships between some intervals or points, find one or more scenarios that are consistent with the information provided, and find all the feasible relations between every pair of interval or points. Expand

A combinatorial approach to temporal reasoning

- Computer Science
- Proceedings of the 5th Jerusalem Conference on Information Technology, 1990. 'Next Decade in Information Technology'
- 1990

It is described how representation structures for counting the number of endpoint sequences associated with a given interval order or interval graph can be used in temporal reasoning. Efficient… Expand

Approximation Algorithms for Temporal Reasoning

- Mathematics, Computer Science
- IJCAI
- 1989

New approximation algorithms for temporal relations between intervals introduced by James Allen are examined, their effectiveness is examined, and under what conditions the algorithms are exact are determined. Expand

Constraint Propagation Algorithms for Temporal Reasoning

- Computer Science
- AAAI
- 1986

Computing the consequences of temporal assertions is shown to be computationally intractable in the interval-based representation, but not in the point-based one, but a fragment of the interval language can be expressed using the point language and benefits from the tractability of the latter. Expand

Interval graphs and related topics

- Computer Science, Mathematics
- Discret. Math.
- 1985

A more general paradigm for studying various classes of graphs is suggested which can be described as follows: when 9 is allowed to be an arbitrary family of sets, the class obtained as intersection graphs is all undirected graphs. Expand

Counting endpoint sequences for interval orders and interval graphs

- Computer Science, Mathematics
- Discret. Math.
- 1993

Efficient algorithms for counting the number of endpoint sequences representing a given interval graph or interval order are designed and analyzed based on the construction of a suitable tree data structure to represent multiple solutions. Expand

An Incremental Linear-Time Algorithm for Recognizing Interval Graphs

- Mathematics, Computer Science
- SIAM J. Comput.
- 1989

This paper presents a much simpler algorithm using a related, but much more informative tree representation of interval graphs. Expand

The complexity of satisfiability problems

- Computer Science, Mathematics
- STOC
- 1978

An infinite class of satisfiability problems is considered which contains these two particular problems as special cases, and it is shown that every member of this class is either polynomial-time decidable or NP-complete. Expand

Interval graphs and interval orders

- Computer Science, Mathematics
- Discret. Math.
- 1985

Two invariants of the family of interval orders that agree with an interval graph are established, namely magnitude, which affects end-point placements, and the property of having the lengths of all representing intervals between specified bounds. Expand

Temporal query processing with indefinite information

- Computer Science
- Artif. Intell. Medicine
- 1991

This paper adopts Allen's influential interval algebra framework for representing temporal information and shows that when the representation language is sufficiently restricted it can develop efficient algorithms for answering interesting classes of queries. Expand