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Dimension and algorithms

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Orders, Algorithms, and Applications (ORDAL 1994)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 831))

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Abstract

Dimension is one of the most heavily studied parameters of partial orders, and many beautiful results have been obtained. However, our knowledge of algorithms for problems on dimension is in a surprisingly primitive state. In this paper, we point to natural problems on dimension which have not been studied, and survey known results in the area.

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References

  1. Blum, A., An O(n 4) approximation algorithm for 3-coloring (and improved approximation algorithms for k-coloring), Proceedings of the 21st Annual ACM Symposium on Theory of Computation, 1991, pp. 535–542.

    Google Scholar 

  2. Buer, H. and R.H. Möhring, A fast algorithm for the decomposition of graphs and posets, Mat. Operations research 8, 1983, pp. 170–184.

    Google Scholar 

  3. Bodlaender, H., T. Kloks and D. Kratsch, Treewidth and pathwidth of permutation graphs, Proceedings of the 20 th International Colloquium on Automata, Languages and Programming, pp. 114–125, Springer Verlag, Lecture Notes in Computer Science, vol. 700, 1993.

    Google Scholar 

  4. Bouchitte, V. and M. Habib, On the greedy dimension of a partial order, Order 1, 1985, pp. 219–224.

    Google Scholar 

  5. Bouchitte, V. and M. Habib, The calculation of invariants for ordered sets, Algorithms and Order, I. Rival (ed.), NATO Series C-Vol 255, Kluwer Academic Publishers, 1989, pp. 231–279.

    Google Scholar 

  6. Brandstädt, A., Special graph classes — a survey, Schriftenreihe des Fachbereichs Mathematik, SM-DU-199, Universität-Gesamthochschule Duisburg, 1991.

    Google Scholar 

  7. Brandstädt, A., On improved time bounds for permutation graph algorithms, Proceedings of the 18th International Workshop on Graph-Theoretic Concepts in Computer science, 1992.

    Google Scholar 

  8. Coppersmith, D. and S. Winograd, Matrix multiplication via arithmetic progressions, Proceedings of the 19th Annual IEEE Symposium on the Foundations of Computer Science, 1987, pp. 1–6.

    Google Scholar 

  9. Cournier, A. and M. Habib, A new linear algorithm for modular decomposition, Research Report 94033, LIRMM, Universite Montpellier 1994.

    Google Scholar 

  10. Deogun, J. S., T. Kloks, D. Kratsch and H. Müller, On vertex ranking for permutation and other graphs, Computing Science Notes 93/30, Eindhoven University of Technology, Eindhoven, The Netherlands, 1993. To appear in: Proceedings of the 11 th Annual Symposium on Theoretical Aspects of Computer Science, 1994.

    Google Scholar 

  11. Dushnik, B. and E. Miller, Partially ordered sets, American Journal of Mathematics 63, (1941), pp. 600–610.

    Google Scholar 

  12. Eschen, E. and J. Spinrad, An O(n 2) algorithm for circular-arc graph recognition, Proceedings of the 4th Annual ACM-SIAM Symposium on Discrete Algorithms, 1993, pp. 128–137.

    Google Scholar 

  13. Even, S., A. Pnueli, A. Lempel, Permutation graphs and transitive graphs, Journal of the ACM 19, 1972, pp. 400–410.

    Google Scholar 

  14. Garey, M.R. and D.S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness, W.H. Freeman, San Francisco, 1979.

    Google Scholar 

  15. Gilmore, P.C. and A.J. Hoffman, A characterization of comparability graphs and interval graphs, Canad. J. Math. 16, 1964, pp. 539–548.

    Google Scholar 

  16. Golumbic, M. C., Algorithmic Graph Theory and Perfect Graphs, Academic Press, New York, 1980.

    Google Scholar 

  17. Golumbic, M. C., D. Rotem, J. Urrutia, Comparability graphs and intersection graphs, Discrete Mathematics 43, 1983, pp. 37–46.

    Google Scholar 

  18. Grötschel, M., L. Lovasz, A. Schrijver, The ellipsoid method and its consequences in combinatorial optimization, Combinatorica 1, 1981, 169–197.

    Google Scholar 

  19. Habib, M. and M.C. Maurer, On the X-Join decomposition for undirected graphs, Appl. Disc. Math 3, 1979, pp. 198–207.

    Google Scholar 

  20. Habib, M. and R. H. Möhring, Treewidth of cocomparability graphs and a new order-theoretic parameter, Technical Report 336/1992, Technische Universität Berlin, 1992.

    Google Scholar 

  21. Hershberger, J., Finding the visibility graph of a polygon in time proportional to its size, Proceedings of the 3d Annual Symposium on Computational Geometry, 1987, pp. 11–20.

    Google Scholar 

  22. Kannan, S., M. Naor, S. Rudich, Implicit representation of graphs, SIAM Journal on discrete Math 5, 1992, pp. 596–603.

    Google Scholar 

  23. Kierstead, H. and W.T. Trotter, The dimension of a cycle-free partial order, Order 9, 1992, 103–110.

    Google Scholar 

  24. Kloks, T., Treewidth, Ph.D. Thesis, Utrecht University, The Netherlands, 1993.

    Google Scholar 

  25. Kloks, T., H. Bodlaender, H. Müller and D. Kratsch, Computing treewidth and minimum fill-in: all you need are the minimal separators, Proceedings of the First Annual European Symposium on Algorithms, pp. 260–271, Springer Verlag, Lecture Notes in Computer Science, Vol. 726, 1993.

    Google Scholar 

  26. Kloks, T. and D. Kratsch, Treewidth of chordal bipartite graphs, Proceedings of the 10 th Annual Symposium on Theoretical Aspects of Computer Science, pp. 80–89, Springer-Verlag, Lecture Notes in Computer Science, Vol. 665, 1993.

    Google Scholar 

  27. Kloks, T. and D. Kratsch, Finding all minimal separators of a graph, Computing Science Note, 93/27, Eindhoven University of Technology, Eindhoven, The Netherlands, (1993). To appear in: Proceedings of the 11 th Annual Symposium on Theoretical Aspects of Computer Science, 1994.

    Google Scholar 

  28. Kloks, T., D. Kratsch, J. Spinrad, Treewidth and pathwidth of cocomparability graphs of bounded dimension, submitted for publication, 1994.

    Google Scholar 

  29. Ma, T.-H. and J. Spinrad, Transitive closure for restricted classes of partial orders, Order 8, 1991, pp. 175–183.

    Google Scholar 

  30. Ma, T.-H. and J. Spinrad, Cycle-free partial orders and chordal comparability graphs, Order 8, 1991, pp. 49–61.

    Google Scholar 

  31. Ma, T.-H. and J. Spinrad, An O(n 2) algorithm for two-chain cover and related problems, Proceedings of the 2d Annual ACM-SIAM Symposium on Discrete Algorithms, 1991, pp. 363–372.

    Google Scholar 

  32. Maurer, S. and A. Ralston, Discrete Algorithmic Mathematics, Addison-Wesley, Reading MA, 1991.

    Google Scholar 

  33. McConnell, R. and J. Spinrad, Linear-time modular decomposition and efficient transitive orientation of comparability graphs, Proceedings of the 5th Annual ACM-SIAM Symposium on discrete Algorithms, 1994, pp. 536–545

    Google Scholar 

  34. Möhring, R. H., Computationally Tractable Classes of Ordered Sets, Algorithms and Order, I. Rival (ed.), NATO Series C — Vol 255, Kluwer Academic Publishers, 1989, pp.105–193.

    Google Scholar 

  35. Möhring, R. H. and F.J. Radermacher, Substitution decomposition for discrete structures and connections with combinatorial optimization, Annals of Discrete Mathematics 19, 1984, pp. 257–356

    Google Scholar 

  36. Möhring, R. H., Triangulating graphs without asteroidal triples, manuscript, TU Berlin, November 1993.

    Google Scholar 

  37. Möhring, R.H., On the Distribution of Locally undecomposable relations and independent systems, Methods Oper. Research 42, 1981, pp. 33–48.

    Google Scholar 

  38. Müller, H., Recognizing interval digraphs and bi-interval graphs in polynomial time, FSU Jena, Germany, manuscript, December 1993.

    Google Scholar 

  39. Muller, J., Local Structure of Graphs, Ph.D. Thesis, Georgia Institute of Technology, 1988.

    Google Scholar 

  40. Muller, J.H. and J.P. Spinrad, Incremental modular decomposition, Journal of the ACM 36, 1989, pp. 1–19.

    Google Scholar 

  41. Overmars, M., Efficient data structures for range searching on a grid, Journal of Algorithms 9, 1988, pp. 254–275.

    Google Scholar 

  42. Preparata, F. and M. Shamos, Computational Geometry, an Introduction, Springer-Verlag, New York, 1985.

    Google Scholar 

  43. Rabinovitch, I., The dimension of semiorders, Journal of Combinatorial Theory A 25 1978, pp. 50–61.

    Google Scholar 

  44. , Rival, I. (ed.), Algorithms and Order, NATO Series C — Vol 255, Kluwer Academic Publishers, 1989.

    Google Scholar 

  45. Roberts, F.S., Graph Theory and its Applications to Problems of Society, SIAM, Philadelphia, 1978.

    Google Scholar 

  46. Spinrad, J.P., On comparability and permutation Graphs, SIAM Journal on Computing 14, 1985, pp.658–670.

    Google Scholar 

  47. Spinrad, J., Nonredundant Ones and Gamma-free Matrices, SIAM Journal on Discrete Math, to appear.

    Google Scholar 

  48. Steiner, G. and L. Stewart, A linear time algorithm to find the jump number of 2-dimensional bipartite partial orders, Order 3, 1987, pp. 359–367.

    Google Scholar 

  49. Trotter, W. T., Combinatorics and Partially Ordered Sets: Dimension Theory, The John Hopkins University Press, Baltimore, Maryland, 1992.

    Google Scholar 

  50. van Emde Boas, P., Preserving order in a forest in less than logarithmic time, Proceedings of the 16th Annual Symposium on Foundations of Computer Science, 1975, pp. 75–84.

    Google Scholar 

  51. Wigderson, A., Improving the performance guarantee for approximate graph coloring, Journal of the ACM 30, 1983, pp. 729–735

    Google Scholar 

  52. Yannakakis, M., The complexity of the partial order dimension problem, SIAM J. Alg. Discrete Methods 3, 1982, pp. 351–358.

    Google Scholar 

  53. Yu, M., L. Tseng, S. Chang, Sequential and parallel algorithms for the maximum-weight independent set problem on permutation graphs, Information Processing Letters 46, 1993, 7–11.

    Google Scholar 

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Authors and Affiliations

  1. Department of Computer Science, Vanderbilt University, 37235, Nashville, TN, USA

    J. Spinrad

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  1. J. Spinrad
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Vincent Bouchitté Michel Morvan

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© 1994 Springer-Verlag Berlin Heidelberg

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Spinrad, J. (1994). Dimension and algorithms. In: Bouchitté, V., Morvan, M. (eds) Orders, Algorithms, and Applications. ORDAL 1994. Lecture Notes in Computer Science, vol 831. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0019425

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  • DOI: https://doi.org/10.1007/BFb0019425

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  • Print ISBN: 978-3-540-58274-8

  • Online ISBN: 978-3-540-48597-1

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