Document Open Access Logo

Binary Constraint Trees and Structured Decomposability

Author Petr Kučera

PDF
Thumbnail PDF

File

LIPIcs.CP.2023.22.pdf
  • Filesize: 0.8 MB
  • 19 pages

Document Identifiers

Author Details

Petr Kučera
  • Department of Theoretical Computer Science and Mathematical Logic, Faculty of Mathematics and Physics, Charles University, Prague, Czech Republic

Cite AsGet BibTex

Petr Kučera. Binary Constraint Trees and Structured Decomposability. In 29th International Conference on Principles and Practice of Constraint Programming (CP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 280, pp. 22:1-22:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.CP.2023.22

Abstract

A binary constraint tree (BCT, Wang and Yap 2022) is a normalized binary CSP whose constraint graph is a tree. A BCT constraint is a constraint represented with a BCT where some of the variables may be hidden (i.e. existentially quantified and used only for internal representation). Structured decomposable negation normal forms (SDNNF) were introduced by Pipatsrisawat and Darwiche (2008) as a restriction of decomposable negation normal forms (DNNF). Both DNNFs and SDNNFs were studied in the area of knowledge compilation. In this paper we show that the BCT constraints are polynomially equivalent to SDNNFs. In particular, a BCT constraint can be represented with an SDNNF of polynomial size and, on the other hand, a constraint that can be represented with an SDNNF, can be represented as a BCT constraint of polynomial size. This generalizes the result of Wang and Yap (2022) that shows that a multivalued decision diagram (MDD) can be represented with a BCT . Moreover, our result provides a full characterization of binary constraint trees using a language that is well studied in the area of knowledge compilation. It was shown by Wang and Yap (2023) that a CSP on n variables of domain sizes bounded by d that has treewidth k can be encoded as a BCT on O(n) variables with domain sizes O(d^{k+1}). We provide an alternative reduction for the case of binary CSPs. This allows us to compile any binary CSP to an SDNNF of size that is parameterized by d and k.

Subject Classification

ACM Subject Classification
  • Theory of computation → Automated reasoning
  • Theory of computation → Constraint and logic programming
Keywords
  • Binary CSP
  • Binary Constraint Tree
  • Structured Decomposability
  • Strucured DNNF
  • Polynomial Equivalence

Metrics

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

References

  1. Ignasi Abío, Graeme Gange, Valentin Mayer-Eichberger, and Peter J. Stuckey. On CNF encodings of decision diagrams. In Claude-Guy Quimper, editor, Integration of AI and OR Techniques in Constraint Programming, pages 1-17, Cham, 2016. Springer International Publishing. Google Scholar
  2. Christian Bessiere and Jean-Charles Régin. Arc consistency for general constraint networks: preliminary results. In IJCAI (1), pages 398-404, 1997. Google Scholar
  3. Simone Bova, Florent Capelli, Stefan Mengel, and Friedrich Slivovsky. On compiling CNFs into structured deterministic DNNFs. In Marijn Heule and Sean Weaver, editors, Theory and Applications of Satisfiability Testing - SAT 2015, pages 199-214, Cham, 2015. Springer International Publishing. Google Scholar
  4. Simone Bova, Florent Capelli, Stefan Mengel, and Friedrich Slivovsky. Knowledge compilation meets communication complexity. In Proceedings of the Twenty-Fifth International Joint Conference on Artificial Intelligence, IJCAI'16, pages 1008-1014. AAAI Press, 2016. URL: http://dl.acm.org/citation.cfm?id=3060621.3060761.
  5. Kenil C. Cheng and Roland H. Yap. Maintaining generalized arc consistency on ad hoc r-ary constraints. In Proceedings of the 14th International Conference on Principles and Practice of Constraint Programming, CP '08, pages 509-523, Berlin, Heidelberg, 2008. Springer-Verlag. URL: https://doi.org/10.1007/978-3-540-85958-1_34.
  6. Kenil C. Cheng and Roland H. Yap. An mdd-based generalized arc consistency algorithm for positive and negative table constraints and some global constraints. Constraints, 15(2):265-304, April 2010. URL: https://doi.org/10.1007/s10601-009-9087-y.
  7. Adnan Darwiche. Compiling knowledge into decomposable negation normal form. In Proceedings of the 16th International Joint Conference on Artifical Intelligence - Volume 1, IJCAI'99, pages 284-289, San Francisco, CA, USA, 1999. Morgan Kaufmann Publishers Inc. URL: http://dl.acm.org/citation.cfm?id=1624218.1624260.
  8. Adnan Darwiche. Decomposable negation normal form. J. ACM, 48(4):608-647, July 2001. URL: https://doi.org/10.1145/502090.502091.
  9. Adnan Darwiche. A compiler for deterministic, decomposable negation normal form. In Eighteenth National Conference on Artificial Intelligence, pages 627-634, Menlo Park, CA, USA, 2002. American Association for Artificial Intelligence. URL: http://dl.acm.org/citation.cfm?id=777092.777189.
  10. Adnan Darwiche and Pierre Marquis. A knowledge compilation map. Journal of Artificial Intelligence Research, 17:229-264, 2002. Google Scholar
  11. Rina Dechter and Judea Pearl. Tree clustering for constraint networks. Artificial Intelligence, 38(3):353-366, April 1989. URL: https://doi.org/10.1016/0004-3702(89)90037-4.
  12. Eugene C. Freuder. A sufficient condition for backtrack-bounded search. Journal of the ACM, 32(4):755-761, October 1985. URL: https://doi.org/10.1145/4221.4225.
  13. Eugene C. Freuder. Complexity of K-tree structured constraint satisfaction problems. In Proceedings of the eighth National conference on Artificial intelligence - Volume 1, AAAI'90, pages 4-9, Boston, Massachusetts, 1990. AAAI Press. Google Scholar
  14. Graeme Gange and Peter J. Stuckey. Explaining propagators for s-DNNF circuits. In Nicolas Beldiceanu, Narendra Jussien, and Éric Pinson, editors, Integration of AI and OR Techniques in Contraint Programming for Combinatorial Optimization Problems, pages 195-210. Springer Berlin Heidelberg, 2012. Google Scholar
  15. Ton Kloks. Treewidth, Computations and Approximations, volume 842 of Lecture Notes in Computer Science. Springer, 1994. URL: https://doi.org/10.1007/BFb0045375.
  16. Petr Kučera and Petr Savický. Propagation complete encodings of smooth DNNF theories. Constraints, 27(3):327-359, July 2022. URL: https://doi.org/10.1007/s10601-022-09331-2.
  17. Christophe Lecoutre. STR2: optimized simple tabular reduction for table constraints. Constraints, 16(4):341-371, October 2011. URL: https://doi.org/10.1007/s10601-011-9107-6.
  18. Christophe Lecoutre, Chavalit Likitvivatanavong, and Roland H. C. Yap. STR3: A path-optimal filtering algorithm for table constraints. Artificial Intelligence, 220:1-27, March 2015. URL: https://doi.org/10.1016/j.artint.2014.12.002.
  19. Robert Mateescu and Rina Dechter. Compiling Constraint Networks into AND/OR Multi-valued Decision Diagrams (AOMDDs). In Frédéric Benhamou, editor, Principles and Practice of Constraint Programming - CP 2006, Lecture Notes in Computer Science, pages 329-343, Berlin, Heidelberg, 2006. Springer. URL: https://doi.org/10.1007/11889205_25.
  20. Knot Pipatsrisawat and Adnan Darwiche. New compilation languages based on structured decomposability. In Proceedings of the 23rd National Conference on Artificial Intelligence - Volume 1, AAAI'08, pages 517-522. AAAI Press, 2008. URL: http://dl.acm.org/citation.cfm?id=1619995.1620079.
  21. Knot Pipatsrisawat and Adnan Darwiche. Top-down algorithms for constructing structured DNNF: Theoretical and practical implications. In Proceedings of the 2010 Conference on ECAI 2010: 19th European Conference on Artificial Intelligence, pages 3-8, Amsterdam, The Netherlands, The Netherlands, 2010. IOS Press. URL: http://dl.acm.org/citation.cfm?id=1860967.1860970.
  22. Francesca Rossi, Charles J. Petrie, and Vasant Dhar. On the equivalence of constraint satisfaction problems. In Proceedings of the 9th European Conference on Artificial Intelligence, ECAI'90, pages 550-556, USA, 1990. Pitman Publishing, Inc. Google Scholar
  23. Andy Shih, Guy Van den Broeck, Paul Beame, and Antoine Amarilli. Smoothing structured decomposable circuits. In H. Wallach, H. Larochelle, A. Beygelzimer, F. dquotesingle Alché-Buc, E. Fox, and R. Garnett, editors, Advances in Neural Information Processing Systems, volume 32. Curran Associates, Inc., 2019. URL: https://proceedings.neurips.cc/paper/2019/file/940392f5f32a7ade1cc201767cf83e31-Paper.pdf.
  24. Kostas Stergiou and Toby Walsh. Encodings of non-binary constraint satisfaction problems. In Proceedings of the Sixteenth National Conference on Artificial Intelligence and the Eleventh Innovative Applications of Artificial Intelligence Conference Innovative Applications of Artificial Intelligence, AAAI '99/IAAI '99, pages 163-168, USA, 1999. American Association for Artificial Intelligence. Google Scholar
  25. Ruiwei Wang and Roland H. C. Yap. Bipartite encoding: A new binary encoding for solving non-binary CSPs. In Proceedings of the Twenty-Ninth International Joint Conference on Artificial Intelligence, IJCAI'20, Yokohama, Yokohama, Japan, 2021. Google Scholar
  26. Ruiwei Wang and Roland H. C. Yap. CNF Encodings of Binary Constraint Trees. In Christine Solnon, editor, 28th International Conference on Principles and Practice of Constraint Programming (CP 2022), volume 235 of Leibniz International Proceedings in Informatics (LIPIcs), pages 40:1-40:19, Dagstuhl, Germany, 2022. Schloss Dagstuhl – Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.CP.2022.40.
  27. Ruiwei Wang and Roland H.C. Yap. Encoding multi-valued decision diagram constraints as binary constraint trees. Proceedings of the AAAI Conference on Artificial Intelligence, 36(4):3850-3858, June 2022. URL: https://doi.org/10.1609/aaai.v36i4.20300.
  28. Ruiwei Wang and Roland H.C. Yap. The expressive power of ad-hoc constraints for modelling CSPs. Proceedings of the AAAI Conference on Artificial Intelligence, 37(4):4104-4114, June 2023. URL: https://doi.org/10.1609/aaai.v37i4.25526.
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