Document Open Access Logo

Exact Exponential Algorithms for Two Poset Problems

Author László Kozma

PDF
Thumbnail PDF

File

LIPIcs.SWAT.2020.30.pdf
  • Filesize: 0.49 MB
  • 15 pages

Document Identifiers

Author Details

László Kozma
  • Freie Universität Berlin, Institute of Computer Science, Germany

Funding

  • Kozma, László: Research supported by DFG grant KO 6140/1-1.

Cite AsGet BibTex

László Kozma. Exact Exponential Algorithms for Two Poset Problems. In 17th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 162, pp. 30:1-30:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
https://doi.org/10.4230/LIPIcs.SWAT.2020.30

Abstract

Partially ordered sets (posets) are fundamental combinatorial objects with important applications in computer science. Perhaps the most natural algorithmic task, given a size-n poset, is to compute its number of linear extensions. In 1991 Brightwell and Winkler showed this problem to be #P-hard. In spite of extensive research, the fastest known algorithm is still the straightforward O(n 2ⁿ)-time dynamic programming (an adaptation of the Bellman-Held-Karp algorithm for the TSP). Very recently, Dittmer and Pak showed that the problem remains #P-hard for two-dimensional posets, and no algorithm was known to break the 2ⁿ-barrier even in this special case. The question of whether the two-dimensional problem is easier than the general case was raised decades ago by Möhring, Felsner and Wernisch, and others. In this paper we show that the number of linear extensions of a two-dimensional poset can be computed in time O(1.8286ⁿ). The related jump number problem asks for a linear extension of a poset, minimizing the number of neighboring incomparable pairs. The problem has applications in scheduling, and has been widely studied. In 1981 Pulleyblank showed it to be NP-complete. We show that the jump number problem can be solved (in arbitrary posets) in time O(1.824ⁿ). This improves (slightly) the previous best bound of Kratsch and Kratsch.

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
Keywords
  • poset
  • linear extension
  • jump number
  • exponential time

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

References

  1. P. D. Amer, C. Chassot, T. J. Connolly, M. Diaz, and P. Conrad. Partial-order transport service for multimedia and other applications. IEEE/ACM Transactions on Networking, 2(5):440-456, October 1994. URL: https://doi.org/10.1109/90.336326.
  2. KA Baker, Peter C Fishburn, and Fred S Roberts. Partial orders of dimension 2, interval orders, and interval graphs, 1970. Google Scholar
  3. Richard Bellman. Dynamic programming treatment of the travelling salesman problem. J. Assoc. Comput. Mach., 9:61-63, 1962. Google Scholar
  4. G. R. Brightwell, S. Felsner, and W. T. Trotter. Balancing pairs and the cross product conjecture. Order, 12(4):327-349, December 1995. URL: https://doi.org/10.1007/BF01110378.
  5. Graham Brightwell. Balanced pairs in partial orders. Discrete Mathematics, 201(1):25-52, 1999. URL: https://doi.org/10.1016/S0012-365X(98)00311-2.
  6. Graham Brightwell and Peter Winkler. Counting linear extensions. Order, 8(3):225-242, September 1991. URL: https://doi.org/10.1007/BF00383444.
  7. Russ Bubley and Martin Dyer. Faster random generation of linear extensions. Discrete Mathematics, 201(1):81-88, 1999. URL: https://doi.org/10.1016/S0012-365X(98)00333-1.
  8. Stéphan Ceroi. A weighted version of the jump number problem on two-dimensional orders is np-complete. Order, 20(1):1-11, 2003. Google Scholar
  9. T.M. Cover and J.A. Thomas. Elements of Information Theory. A Wiley-Interscience publication. Wiley, 2006. Google Scholar
  10. Marek Cygan, Holger Dell, Daniel Lokshtanov, Dániel Marx, Jesper Nederlof, Yoshio Okamoto, Ramamohan Paturi, Saket Saurabh, and Magnus Wahlström. On problems as hard as cnf-sat. ACM Trans. Algorithms, 12(3):41:1-41:24, May 2016. URL: https://doi.org/10.1145/2925416.
  11. Karel De Loof, Hans De Meyer, and Bernard De Baets. Exploiting the lattice of ideals representation of a poset. Fundam. Inf., 71(2,3):309-321, February 2006. Google Scholar
  12. Samuel Dittmer and Igor Pak. Counting linear extensions of restricted posets, 2018. URL: http://arxiv.org/abs/1802.06312.
  13. Ben Dushnik and E. W. Miller. Partially ordered sets. American Journal of Mathematics, 63(3):600-610, 1941. Google Scholar
  14. Martin Dyer, Alan Frieze, and Ravi Kannan. A random polynomial-time algorithm for approximating the volume of convex bodies. J. ACM, 38(1):1-17, January 1991. URL: https://doi.org/10.1145/102782.102783.
  15. E. Eiben, R. Ganian, K. Kangas, and S. Ordyniak. Counting linear extensions: Parameterizations by treewidth. Algorithmica, 81(4):1657-1683, April 2019. URL: https://doi.org/10.1007/s00453-018-0496-4.
  16. Stefan Felsner and Thibault Manneville. Linear extensions of n-free orders. Order, 32(2):147-155, 2015. URL: https://doi.org/10.1007/s11083-014-9321-0.
  17. Stefan Felsner and Lorenz Wernisch. Markov chains for linear extensions, the two-dimensional case. In Proceedings of the Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, 5-7 January 1997, New Orleans, Louisiana, USA., pages 239-247, 1997. URL: http://dl.acm.org/citation.cfm?id=314161.314262.
  18. Peter C. Fishburn and William V. Gehrlein. A comparative analysis of methods for constructing weak orders from partial orders. The Journal of Mathematical Sociology, 4(1):93-102, 1975. URL: https://doi.org/10.1080/0022250X.1975.9989846.
  19. R. L. Graham, M. Grötschel, and L. Lovász, editors. Handbook of Combinatorics (Vol. 1). MIT Press, Cambridge, MA, USA, 1995. Google Scholar
  20. Michael Held and Richard M. Karp. A dynamic programming approach to sequencing problems. J. Soc. Indust. Appl. Math., 10:196-210, 1962. Google Scholar
  21. Russell Impagliazzo, Ramamohan Paturi, and Francis Zane. Which problems have strongly exponential complexity? J. Comput. Syst. Sci., 63(4):512-530, December 2001. URL: https://doi.org/10.1006/jcss.2001.1774.
  22. Gwenaël Joret, Piotr Micek, and Veit Wiechert. Sparsity and dimension. Combinatorica, 38(5):1129-1148, October 2018. URL: https://doi.org/10.1007/s00493-017-3638-4.
  23. Jeff Kahn and Michael Saks. Balancing poset extensions. Order, 1(2):113-126, June 1984. URL: https://doi.org/10.1007/BF00565647.
  24. Kustaa Kangas, Teemu Hankala, Teppo Mikael Niinimäki, and Mikko Koivisto. Counting linear extensions of sparse posets. In IJCAI 2016, pages 603-609, 2016. URL: http://www.ijcai.org/Abstract/16/092.
  25. Kustaa Kangas, Mikko Koivisto, and Sami Salonen. A faster tree-decomposition based algorithm for counting linear extensions. In IPEC 2018, pages 5:1-5:13, 2018. URL: https://doi.org/10.4230/LIPIcs.IPEC.2018.5.
  26. Donald E. Knuth and Jayme Luiz Szwarcfiter. A structured program to generate all topological sorting arrangements. Inf. Process. Lett., 2(6):153-157, 1974. URL: https://doi.org/10.1016/0020-0190(74)90001-5.
  27. Dieter Kratsch and Stefan Kratsch. The jump number problem: Exact and parameterized. In IPEC 2013, pages 230-242, 2013. URL: https://doi.org/10.1007/978-3-319-03898-8_20.
  28. Nathan Linial. Hard enumeration problems in geometry and combinatorics. SIAM J. Algebraic Discrete Methods, 7(2):331-335, April 1986. URL: https://doi.org/10.1137/0607036.
  29. L. Lovász. An Algorithmic Theory of Numbers, Graphs, and Convexity. CBMS-NSF Regional Conference Series in Applied Mathematics. SIAM, 1986. Google Scholar
  30. Thomas Lukasiewicz, Maria Vanina Martinez, and Gerardo I. Simari. Probabilistic preference logic networks. In ECAI 2014, pages 561-566, 2014. URL: https://doi.org/10.3233/978-1-61499-419-0-561.
  31. Heikki Mannila and Christopher Meek. Global partial orders from sequential data. In ACM SIGKDD 2000, pages 161-168, 2000. URL: https://doi.org/10.1145/347090.347122.
  32. Jason Morton, Lior Pachter, Anne Shiu, Bernd Sturmfels, and Oliver Wienand. Convex rank tests and semigraphoids. SIAM J. Discrete Math., 23(3):1117-1134, 2009. URL: https://doi.org/10.1137/080715822.
  33. Rolf Möhring. Computationally Tractable Classes of Ordered Sets, pages 105-193. Springer, January 1989. URL: https://doi.org/10.1007/978-94-009-2639-4_4.
  34. Teppo Mikael Niinimäki and Mikko Koivisto. Annealed importance sampling for structure learning in bayesian networks. In IJCAI 2013, Proceedings of the 23rd International Joint Conference on Artificial Intelligence, Beijing, China, August 3-9, 2013, pages 1579-1585, 2013. URL: http://www.aaai.org/ocs/index.php/IJCAI/IJCAI13/paper/view/6885.
  35. J. Scott Provan and Michael O. Ball. The complexity of counting cuts and of computing the probability that a graph is connected. SIAM J. Comput., 12(4):777-788, 1983. URL: https://doi.org/10.1137/0212053.
  36. William R Pulleyblank. On minimizing setups in precedence constrained scheduling. Universität Bonn. Institut für Ökonometrie und Operations Research, 1981. Google Scholar
  37. I. Rival. Ordered Sets: Proceedings of the NATO Advanced Study Institute held at Banff, Canada, August 28 to September 12, 1981. Nato Science Series C:. Springer Netherlands, 2012. Google Scholar
  38. Alex Scott and David R. Wood. Better bounds for poset dimension and boxicity. CoRR, abs/1804.03271, 2018. URL: http://arxiv.org/abs/1804.03271.
  39. Richard P Stanley. Two poset polytopes. Discrete & Computational Geometry, 1(1):9-23, 1986. Google Scholar
  40. R.P. Stanley. Enumerative Combinatorics:. Cambridge Studies in Advanced Mathematics. Cambridge University Press, 1997. Google Scholar
  41. George Steiner. On estimating the number of order ideals in partial orders, with some applications. Journal of statistical planning and inference, 34(2):281-290, 1993. Google Scholar
  42. George Steiner and Lorna K Stewart. A linear time algorithm to find the jump number of 2-dimensional bipartite partial orders. Order, 3(4):359-367, 1987. Google Scholar
  43. R.E. Tarjan. Data Structures and Network Algorithms. CBMS-NSF Regional Conference Series in Applied Mathematics. Society for Industrial and Applied Mathematics, 1983. Google Scholar
  44. W.T. Trotter. Combinatorics and partially ordered sets: dimension theory. Johns Hopkins Series in the Mathematical Sciences. J. Hopkins University Press, 1992. Google Scholar
  45. Chris S. Wallace, Kevin B. Korb, and Honghua Dai. Causal discovery via mml. In ICML 1996, pages 516-524, San Francisco, CA, USA, 1996. Morgan Kaufmann Publishers Inc. Google Scholar
  46. P. Winkler. Average height in a partially ordered set. Discrete Mathematics, 39(3):337-341, 1982. URL: https://doi.org/10.1016/0012-365X(82)90157-1.
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2024 Schloss Dagstuhl – LZI GmbH Imprint Privacy Contact