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

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. 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 NP-complete

[1]  Frank Ruskey,et al.  Generating Linear Extensions Fast , 1994, SIAM J. Comput..

[2]  Robert H. Halstead,et al.  Computation structures , 1990, MIT electrical engineering and computer science series.

[3]  Martin Charles Golumbic.  Interval graphs and related topics , 1985, Discret. Math..

[4]  Peter van Beek,et al.  Approximation Algorithms for Temporal Reasoning , 1989, IJCAI.

[5]  Kellogg S. Booth,et al.  Testing for the Consecutive Ones Property, Interval Graphs, and Graph Planarity Using PQ-Tree Algorithms , 1976, J. Comput. Syst. Sci..

[6]  Peter C. Fishburn,et al.  Partial orders of dimension 2 , 1972, Networks.

[7]  Ronald L. Rivest,et al.  The Design and Analysis of Computer Algorithms , 1990 .

[8]  Peter van Beek,et al.  Reasoning About Qualitative Temporal Information , 1990, Artif. Intell..

[9]  P. Gilmore,et al.  A Characterization of Comparability Graphs and of Interval Graphs , 1964, Canadian Journal of Mathematics.

[10]  Alfred V. Aho,et al.  The Design and Analysis of Computer Algorithms , 1974 .

[11]  André Bouchet,et al.  Reducing prime graphs and recognizing circle graphs , 1987, Comb..

[12]  P. Fishburn.  Intransitive indifference with unequal indifference intervals , 1970 .

[13]  Henry A. Kautz,et al.  Constraint Propagation Algorithms for Temporal Reasoning , 1986, AAAI.

[14]  R. Möhring.  Algorithmic graph theory and perfect graphs , 1986 .

[15]  David G. Kendall,et al.  Some problems and methods in statistical archaeology , 1969 .

[16]  D. R. Fulkerson,et al.  Incidence matrices and interval graphs , 1965 .

[17]  Peter Winkler,et al.  Counting linear extensions is #P-complete , 1991, STOC '91.

[18]  F. Roberts.  Discrete Mathematical Models with Applications to Social, Biological, and Environmental Problems. , 1976 .

[19]  Thomas J. Schaefer,et al.  The complexity of satisfiability problems , 1978, STOC.

[20]  Joseph Naor,et al.  Simple and Fast Algorithms for Linear and Integer Programs With Two Variables per Inequality , 1994, SIAM J. Comput..

[21]  Peter C. Fishburn,et al.  Interval graphs and interval orders , 1985, Discret. Math..

[22]  James F. Allen.  Maintaining knowledge about temporal intervals , 1983, CACM.

[23]  D. Kendall.  Incidence matrices, interval graphs and seriation in archeology. , 1969 .

[24]  Mihalis Yannakakis,et al.  Scheduling Interval-Ordered Tasks , 1979, SIAM J. Comput..

[25]  Martin Charles Golumbic,et al.  A combinatorial approach to temporal reasoning , 1990, Proceedings of the 5th Jerusalem Conference on Information Technology, 1990. 'Next Decade in Information Technology'.

[26]  Bernard Nudel,et al.  Consistent-Labeling Problems and Their Algorithms: Expected-Complexities and Theory-Based Heuristics , 1983, Artif. Intell..

[27]  Rolf H. Möhring,et al.  An Incremental Linear-Time Algorithm for Recognizing Interval Graphs , 1989, SIAM J. Comput..

[28]  Robert E. Tarjan,et al.  Depth-First Search and Linear Graph Algorithms , 1972, SIAM J. Comput..

[29]  Wen-Lian Hsu,et al.  Recognizing circle graphs in polynomial time , 1985, 26th Annual Symposium on Foundations of Computer Science (sfcs 1985).

[30]  Peter van Beek,et al.  Temporal query processing with indefinite information , 1991, Artif. Intell. Medicine.

[31]  S. Benzer.  ON THE TOPOLOGY OF THE GENETIC FINE STRUCTURE. , 1959, Proceedings of the National Academy of Sciences of the United States of America.

[32]  Klaus Nökel.  Temporally Distributed Symptoms in Technical Diagnosis , 1991, Lecture Notes in Computer Science.

[33]  C. Lekkeikerker,et al.  Representation of a finite graph by a set of intervals on the real line , 1962 .

[34]  Clyde H. Coombs,et al.  On the detection of structure in attitudes and developmental processes. , 1973 .

[35]  A. Carrano.  Establishing the order of human chromosome-specific DNA fragments. , 1988, Basic life sciences.

[36]  Martin Charles Golumbic,et al.  Counting endpoint interval orders and , 2001 .

[37]  Alfred Tarski,et al.  Relational selves as self-affirmational resources , 2008 .

[38]  GolumbicMartin Charles,et al.  Complexity and algorithms for reasoning about time , 1993 .

[39]  Ben Dushnik,et al.  Partially Ordered Sets , 1941 .