SpringerLink
Log in

MaxSAT-Based Postprocessing for Treedepth

  • Conference paper
  • First Online:
Principles and Practice of Constraint Programming (CP 2020)

Part of the book series: Lecture Notes in Computer Science ((LNPSE,volume 12333))

Abstract

Treedepth is an increasingly popular graph invariant. Many NP-hard combinatorial problems can be solved efficiently on graphs of bounded treedepth. Since the exact computation of treedepth is itself NP-hard, recent research has focused on the development of heuristics that compute good upper bounds on the treedepth.

In this paper, we introduce a novel MaxSAT-based approach for improving a heuristically obtained treedepth decomposition. At the core of our approach is an efficient MaxSAT encoding of a weighted generalization of treedepth arising naturally due to subtree contractions. The encoding is applied locally to the given treedepth decomposition to reduce its depth, in conjunction with the collapsing of subtrees. We show the local improvement method’s correctness and provide an extensive experimental evaluation with some encouraging results.

The authors acknowledge the support by the FWF (projects P32441 and W1255) and by the WWTF (project ICT19-065).

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
EUR 29.95
Price includes VAT (Germany)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
EUR 117.69
Price includes VAT (Germany)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
EUR 160.49
Price includes VAT (Germany)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free ship** worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

Notes

  1. 1.

    https://pacechallenge.org/2020.

  2. 2.

    https://www.labri.fr/perso/lsimon/glucose/.

  3. 3.

    https://maxsat-evaluations.github.io/2019/descriptions.html.

References

  1. Bannach, M., Berndt, S., Ehlers, T.: Jdrasil: a modular library for computing tree decompositions. In: Iliopoulos, C.S., Pissis, S.P., Puglisi, S.J., Raman, R. (eds.) 16th International Symposium on Experimental Algorithms, SEA 2017, London, UK, 21–23 June 2017, vol. 75, pp. 28:1–28:21. LIPIcs, Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik (2017)

    Google Scholar 

  2. Berg, J., Järvisalo, M.: SAT-based approaches to treewidth computation: an evaluation. In: 26th IEEE International Conference on Tools with Artificial Intelligence, ICTAI 2014, Limassol, Cyprus, 10–12 November 2014, pp. 328–335. IEEE Computer Society (2014)

    Google Scholar 

  3. Dechter, R., Mateescu, R.: AND/OR search spaces for graphical models. Artif. Intell. 171(2–3), 73–106 (2007)

    Article  MathSciNet  Google Scholar 

  4. Fichte, J.K., Hecher, M., Lodha, N., Szeider, S.: An SMT approach to fractional hypertree width. In: Hooker, J. (ed.) CP 2018. LNCS, vol. 11008, pp. 109–127. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-98334-9_8

    Chapter  Google Scholar 

  5. Fichte, J.K., Lodha, N., Szeider, S.: SAT-based local improvement for finding tree decompositions of small width. In: Gaspers, S., Walsh, T. (eds.) SAT 2017. LNCS, vol. 10491, pp. 401–411. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-66263-3_25

    Chapter  MATH  Google Scholar 

  6. Fomin, F.V., Giannopoulou, A.C., Pilipczuk, M.: Computing tree-depth faster than 2\({}^{\text{ n }}\). Algorithmica 73(1), 202–216 (2015)

    Article  MathSciNet  Google Scholar 

  7. Freuder, E.C., Quinn, M.J.: Taking advantage of stable sets of variables in constraint satisfaction problems. In: IJCAI, vol. 85, pp. 1076–1078. Citeseer (1985)

    Google Scholar 

  8. Gajarský, J., Hlinený, P.: Faster deciding MSO properties of trees of fixed height, and some consequences. In: D’Souza, D., Kavitha, T., Radhakrishnan, J. (eds.) IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science, FSTTCS 2012, Hyderabad, India, 15–17 December 2012, vol. 18, pp. 112–123. LIPIcs, Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik (2012)

    Google Scholar 

  9. Ganian, R., Lodha, N., Ordyniak, S., Szeider, S.: SAT-encodings for treecut width and treedepth. In: Kobourov, S.G., Meyerhenke, H. (eds.) Proceedings of ALENEX 2019, the 21st Workshop on Algorithm Engineering and Experiments, pp. 117–129. SIAM (2019)

    Google Scholar 

  10. Gutin, G., Jones, M., Wahlström, M.: Structural parameterizations of the mixed Chinese postman problem. In: Bansal, N., Finocchi, I. (eds.) ESA 2015. LNCS, vol. 9294, pp. 668–679. Springer, Heidelberg (2015). https://doi.org/10.1007/978-3-662-48350-3_56

    Chapter  Google Scholar 

  11. Hafsteinsson, H.: Parallel sparse Cholesky factorization. Cornell University, Technical report (1988)

    Google Scholar 

  12. Hagberg, A.A., Schult, D.A., Swart, P.J.: Exploring network structure, dynamics, and function using NetworkX. In: Proceedings of the 7th Python in Science Conference (SciPy 2008), Pasadena, CA, USA, pp. 11–15, August 2008

    Google Scholar 

  13. Heule, M., Szeider, S.: A SAT approach to clique-width. ACM Trans. Comput. Log. 16(3), 24 (2015). https://doi.org/10.1145/2736696

    Article  MathSciNet  MATH  Google Scholar 

  14. Ignatiev, A., Morgado, A., Marques-Silva, J.: PySAT: a Python toolkit for prototy** with SAT Oracles. In: Beyersdorff, O., Wintersteiger, C.M. (eds.) SAT 2018. LNCS, vol. 10929, pp. 428–437. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-94144-8_26

    Chapter  MATH  Google Scholar 

  15. Iwata, Y., Ogasawara, T., Ohsaka, N.: On the power of tree-depth for fully polynomial FPT algorithms. In: Niedermeier, R., Vallée, B. (eds.) 35th Symposium on Theoretical Aspects of Computer Science, STACS 2018, 28 February–3 March 2018, Caen, France, vol. 96, pp. 41:1–41:14. LIPIcs, Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik (2018)

    Google Scholar 

  16. Iyer, A.V., Ratliff, H.D., Vijayan, G.: On a node ranking problem of trees and graphs. Technical reporrt, Georgia Inst of Tech Atlanta Production and Distribution Research Center (1986)

    Google Scholar 

  17. Jégou, P., Terrioux, C.: Hybrid backtracking bounded by tree-decomposition of constraint networks. Artif. Intell. 146(1), 43–75 (2003)

    Article  MathSciNet  Google Scholar 

  18. Jess, J.A.G., Kees, H.G.M.: A data structure for parallel L/U decomposition. IEEE Trans. Comput. 3, 231–239 (1982)

    Article  MathSciNet  Google Scholar 

  19. Kayaaslan, E., Uçar, B.: Reducing elimination tree height for parallel LU factorization of sparse unsymmetric matrices. In: 2014 21st International Conference on High Performance Computing (HiPC), pp. 1–10. IEEE (2014)

    Google Scholar 

  20. Kees, H.G.M.: The organization of circuit analysis on array architectures. Ph.D. thesis, Citeseer (1982)

    Google Scholar 

  21. Liu, J.W.: Reordering sparse matrices for parallel elimination. Parallel Comput. 11(1), 73–91 (1989)

    Article  MathSciNet  Google Scholar 

  22. Liu, J.W.: The role of elimination trees in sparse factorization. SIAM J. Mat. Anal. Appl. 11(1), 134–172 (1990)

    Article  MathSciNet  Google Scholar 

  23. Llewellyn, D.C., Tovey, C., Trick, M.: Local optimization on graphs. Discrete Appl. Math. 23(2), 157–178 (1989)

    Article  MathSciNet  Google Scholar 

  24. Lodha, N., Ordyniak, S., Szeider, S.: A SAT approach to branchwidth. In: Creignou, N., Le Berre, D. (eds.) SAT 2016. LNCS, vol. 9710, pp. 179–195. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-40970-2_12

    Chapter  MATH  Google Scholar 

  25. Lodha, N., Ordyniak, S., Szeider, S.: SAT-encodings for special treewidth and pathwidth. In: Gaspers, S., Walsh, T. (eds.) SAT 2017. LNCS, vol. 10491, pp. 429–445. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-66263-3_27

    Chapter  MATH  Google Scholar 

  26. Lodha, N., Ordyniak, S., Szeider, S.: A SAT approach to branchwidth. ACM Trans. Comput. Log. 20(3), 15:1–15:24 (2019)

    Google Scholar 

  27. Lusseau, D., Schneider, K., Boisseau, O.J., Haase, P., Slooten, E., Dawson, S.M.: The bottlenose dolphin community of doubtful sound features a large proportion of long-lasting associations. Behav. Ecol. Sociobiol. 54(4), 396–405 (2003)

    Article  Google Scholar 

  28. Manne, F.: Reducing the height of an elimination tree through local recorderings. University of Bergen, Department of Informatics (1991)

    Google Scholar 

  29. Nesetril, J., de Mendez, P.O.: Tree-depth, subgraph coloring and homomorphism bounds. Eur. J. Comb. 27(6), 1022–1041 (2006)

    Article  MathSciNet  Google Scholar 

  30. Nešetřil, J., de Mendez, P.O.: Sparsity - Graphs, Structures, and Algorithms. Algorithms and Combinatorics, vol. 28. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-27875-4

    Book  MATH  Google Scholar 

  31. Oelschlägel, T.: Treewidth from Treedepth. Ph.D. thesis, RWTH Aachen University (2014)

    Google Scholar 

  32. Peruvemba Ramaswamy, V., Szeider, S.: aditya95sriram/td-slim: public release, July 2020. Zenodo. https://doi.org/10.5281/zenodo.3946663

  33. Peruvemba Ramaswamy, V., Szeider, S.: Turbocharging treewidth-bounded Bayesian network structure learning (2020). https://arxiv.org/abs/2006.13843

  34. Pieck, J.: Formele definitie van een e-tree. Eindhoven University of Technology: Department of Mathematics: Memorandum 8006 (1980)

    Google Scholar 

  35. Pisinger, D., Ropke, S.: Large neighborhood search. In: Gendreau, M., Potvin, J.Y. (eds.) Handbook of Metaheuristics, pp. 399–419. Springer, Boston (2010). https://doi.org/10.1007/978-1-4419-1665-5_13

    Chapter  Google Scholar 

  36. Pothen, A.: The complexity of optimal elimination trees. Technical report (1988)

    Google Scholar 

  37. Pothen, A., Simon, H.D., Wang, L., Barnard, S.T.: Towards a fast implementation of spectral nested dissection. In: Supercomputing 1992: Proceedings of the 1992 ACM/IEEE Conference on Supercomputing, pp. 42–51. IEEE (1992)

    Google Scholar 

  38. Reidl, F., Rossmanith, P., Villaamil, F.S., Sikdar, S.: A faster parameterized algorithm for treedepth. In: Esparza, J., Fraigniaud, P., Husfeldt, T., Koutsoupias, E. (eds.) ICALP 2014. LNCS, vol. 8572, pp. 931–942. Springer, Heidelberg (2014). https://doi.org/10.1007/978-3-662-43948-7_77

    Chapter  Google Scholar 

  39. Samer, M., Veith, H.: Encoding treewidth into SAT. In: Kullmann, O. (ed.) SAT 2009. LNCS, vol. 5584, pp. 45–50. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-642-02777-2_6

    Chapter  MATH  Google Scholar 

  40. Schidler, A., Szeider, S.: Computing optimal hypertree decompositions. In: Blelloch, G., Finocchi, I. (eds.) Proceedings of ALENEX 2020, the 22nd Workshop on Algorithm Engineering and Experiments, pp. 1–11. SIAM (2020)

    Google Scholar 

  41. Swat, S.: swacisko/pace-2020: first release of ExTREEm, June 2020. Zenodo. https://doi.org/10.5281/zenodo.3873126

  42. Villaamil, F.S.: About treedepth and related notions. Ph.D. thesis, RWTH Aachen University (2017)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Algorithms and Complexity Group, TU Wien, Vienna, Austria

    Vaidyanathan Peruvemba Ramaswamy & Stefan Szeider

Authors
  1. Vaidyanathan Peruvemba Ramaswamy
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Stefan Szeider
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Vaidyanathan Peruvemba Ramaswamy .

Editor information

Editors and Affiliations

  1. Insight Centre for Data Analytics, School for Computer Science and Information Technology, University College Cork, Cork, Ireland

    Helmut Simonis

Rights and permissions

Reprints and permissions

Copyright information

© 2020 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Peruvemba Ramaswamy, V., Szeider, S. (2020). MaxSAT-Based Postprocessing for Treedepth. In: Simonis, H. (eds) Principles and Practice of Constraint Programming. CP 2020. Lecture Notes in Computer Science(), vol 12333. Springer, Cham. https://doi.org/10.1007/978-3-030-58475-7_28

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-58475-7_28

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-58474-0

  • Online ISBN: 978-3-030-58475-7

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation