Skip to main content
SpringerLink
Log in

Positive-Instance Driven Dynamic Programming for Graph Searching

  • Conference paper
  • First Online:
Algorithms and Data Structures (WADS 2019)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11646))

Included in the following conference series:

Abstract

Research on the similarity of a graph to being a tree – called the treewidth of the graph – has seen an enormous rise within the last decade, but a practically fast algorithm for this task has been discovered only recently by Tamaki (ESA 2017). It is based on dynamic programming and makes use of the fact that the number of positive subinstances is typically substantially smaller than the number of all subinstances. Algorithms producing only such subinstances are called positive-instance driven (PID). We give an alternative and intuitive view on this algorithm from the perspective of the corresponding configuration graphs in certain two-player games. This allows us to develop PID-algorithms for a wide range of important graph parameters such as treewidth, pathwidth, and treedepth. We analyse the worst case behaviour of the approach on some well-known graph classes and perform an experimental evaluation on real world and random graphs.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Arnborg, S., Corneil, D.G., Proskurowski, A.: Complexity of finding embeddings in a k-tree. SIAM J. Algebraic Discrete Methods 8(2), 277–284 (1987)

    Article  MathSciNet  Google Scholar 

  2. Bannach, M., Berndt, S., , T.E.: Jdrasil (2017). https://github.com/maxbannach/Jdrasil. Accessed 09 Feb 2019

  3. Bannach, M., Berndt, S.: Practical access to dynamic programming on tree decompositions. In: ESA, pp. 6:1–6:13 (2018)

    Google Scholar 

  4. Bannach, M., Berndt, S., Ehlers, T.: Jdrasil: a modular library for computing tree decompositions. In: SEA, pp. 28:1–28:21 (2017)

    Google Scholar 

  5. Berg, J., Järvisalo, M., Malone, B.: Learning optimal bounded treewidth bayesian networks via maximum satisfiability. In: Artificial Intelligence and Statistics, pp. 86–95 (2014)

    Google Scholar 

  6. Bienstock, D., Seymour, P.D.: Monotonicity in graph searching. J. Algorithms 12(2), 239–245 (1991)

    Article  MathSciNet  Google Scholar 

  7. Bjesse, P., Kukula, J., Damiano, R., Stanion, T., Zhu, Y.: Guiding SAT diagnosis with tree decompositions. In: Giunchiglia, E., Tacchella, A. (eds.) SAT 2003. LNCS, vol. 2919, pp. 315–329. Springer, Heidelberg (2004). https://doi.org/10.1007/978-3-540-24605-3_24

    Chapter  Google Scholar 

  8. Bodlaender, H.L.: A linear-time algorithm for finding tree-decompositions of small treewidth. SIAM J. Comput. 25(6), 1305–1317 (1996)

    Article  MathSciNet  Google Scholar 

  9. Bodlaender, H.L., Kratsch, S., Kreuzen, V.J.C., Kwon, O., Ok, S.: Characterizing width two for variants of treewidth. Discrete Appl. Math. 216, 29–46 (2017)

    Article  MathSciNet  Google Scholar 

  10. Bollobás, B.: Random graphs. In: Bollobás, B. (ed.) Modern Graph Theory, pp. 215–252. Springer, New York (1998). https://doi.org/10.1007/978-1-4612-0619-4_7

    Chapter  MATH  Google Scholar 

  11. Bouchitté, V., Todinca, I.: Listing all potential maximal cliques of a graph. Theor. Comput. Sci. 276(1–2), 17–32 (2002)

    Article  MathSciNet  Google Scholar 

  12. Charwat, G., Woltran, S.: Dynamic programming-based QBF solving. In: QBF, pp. 27–40 (2016)

    Google Scholar 

  13. Courcelle, B.: On the model-checking of monadic second-order formulas with edge set quantifications. Discrete Appl. Math. 160(6), 866–887 (2012)

    Article  MathSciNet  Google Scholar 

  14. Cygan, M., et al.: Parameterized Algorithms. Springer, Berlin Heidelberg (2015). https://doi.org/10.1007/978-3-319-21275-3

    Book  MATH  Google Scholar 

  15. Darwiche, A.: A differential approach to inference in bayesian networks. J. ACM (JACM) 50(3), 280–305 (2003)

    Article  MathSciNet  Google Scholar 

  16. Dell, H., Husfeldt, T., Jansen, B., Kaski, P., Komusiewicz, C., Rosamond, F.: The first parameterized algorithms and computational experiments challenge. In: IPEC 2016, pp. 30:1–30:9 (2017)

    Google Scholar 

  17. Dell, H., Komusiewicz, C., Talmon, N., Weller, M.: The PACE 2017 parameterized algorithms and computational experiments challenge: the second iteration. In: IPEC 2017 (2018)

    Google Scholar 

  18. Eiben, E., Ganian, R., Ordyniak, S.: Small resolution proofs for QBF using dependency treewidth. In: STACS. LIPIcs, vol. 96, pp. 28:1–28:15. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik (2018)

    Google Scholar 

  19. Eisenbrand, F., Hunkenschröder, C., Klein, K.: Faster algorithms for integer programs with block structure. In: ICALP. LIPIcs, vol. 107, pp. 49:1–49:13. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik (2018)

    Google Scholar 

  20. Elidan, G., Gould, S.: Learning bounded treewidth bayesian networks. J. Mach. Learn. Res. 9(Dec), 2699–2731 (2008)

    MathSciNet  MATH  Google Scholar 

  21. Fichte, J.K., Hecher, M., Woltran, S., Zisser, M.: Weighted model counting on the GPU by exploiting small treewidth. In: ESA, pp. 28:1–28:16 (2018)

    Google Scholar 

  22. Fomin, F., Fraigniaud, P., Nisse, N.: Nondeterministic graph searching: from pathwidth to treewidth. Algorithmica 53(3), 358–373 (2009)

    Article  MathSciNet  Google Scholar 

  23. Fomin, F.V., Kratsch, D.: Exact Exponential Algorithms. Texts in Theoretical Computer Science. An EATCS Series. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-16533-7

    Book  MATH  Google Scholar 

  24. Ganian, R., Ordyniak, S.: The complexity landscape of decompositional parameters for ILP. Artif. Intell. 257, 61–71 (2018)

    Article  MathSciNet  Google Scholar 

  25. Ganian, R., Ordyniak, S., Ramanujan, M.S.: Going beyond primal treewidth for (M)ILP. In: AAAI, pp. 815–821. AAAI Press (2017)

    Google Scholar 

  26. Giannopoulou, A., Hunter, P., Thilikos, D.: LIFO-search: a min-max theorem and a searching game for cycle-rank and tree-depth. Discrete Appl. Math. 160(15), 2089–2097 (2012)

    Article  MathSciNet  Google Scholar 

  27. Habet, D., Paris, L., Terrioux, C.: A tree decomposition based approach to solve structured SAT instances. In: ICTAI, pp. 115–122 (2009)

    Google Scholar 

  28. Immerman, N.: Descriptive Complexity. Springer, Heidelberg (1999). https://doi.org/10.1007/978-1-4612-0539-5

    Book  MATH  Google Scholar 

  29. Johnson, D.S., Trick, M.A.: Cliques, coloring, and satisfiability: second DIMACS implementation challenge, 11–13 October 1993, vol. 26. American Mathematical Soc. (1996)

    Google Scholar 

  30. Kahn, A.B.: Topological sorting of large networks. Commun. ACM 5(11), 558–562 (1962)

    Article  Google Scholar 

  31. Karakashian, S., Woodward, R.J., Choueiry, B.Y.: Improving the performance of consistency algorithms by localizing and bolstering propagation in a tree decomposition. In: AAAI (2013)

    Google Scholar 

  32. Kneis, J., Langer, A., Rossmanith, P.: Courcelle’s theorem - a game-theoretic approach. Discrete Optim. 8(4), 568–594 (2011)

    Article  MathSciNet  Google Scholar 

  33. Koster, A.M.C.A., van Hoesel, S.P.M., Kolen, A.W.J.: Solving partial constraint satisfaction problems with tree decomposition. Networks 40(3), 170–180 (2002)

    Article  MathSciNet  Google Scholar 

  34. Koutecký, M., Levin, A., Onn, S.: A parameterized strongly polynomial algorithm for block structured integer programs. In: ICALP. LIPIcs, vol. 107, pp. 85:1–85:14. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik (2018)

    Google Scholar 

  35. LaPaugh, A.: Recontamination does not help to search a graph. J. ACM 40, 224–245 (1993)

    Article  MathSciNet  Google Scholar 

  36. Larisch, L., Salfelder, F.: p17 (2017). https://github.com/freetdi/p17. Accessed 02 Aug 2017

  37. Mazoit, F., Nisse, N.: Monotonicity of non-deterministic graph searching. TCS 399(3), 169–178 (2008)

    Article  MathSciNet  Google Scholar 

  38. Röhrig, H.: Tree decomposition: a feasibility Study. Diploma thesis, Max-Planck-Institut für Informatik in Saarbrücken (1998)

    Google Scholar 

  39. Seymour, P., Thomas, R.: Graph searching and a min-max theorem for tree-width. JCT 58, 22–33 (1993)

    Article  MathSciNet  Google Scholar 

  40. Szeider, S.: On fixed-parameter tractable parameterizations of SAT. In: Giunchiglia, E., Tacchella, A. (eds.) SAT 2003. LNCS, vol. 2919, pp. 188–202. Springer, Heidelberg (2004). https://doi.org/10.1007/978-3-540-24605-3_15

    Chapter  MATH  Google Scholar 

  41. Tamaki, H.: treewidth-exact (2016). https://github.com/TCS-Meiji/treewidth-exact. Accessed 02 Aug 2017

  42. Tamaki, H.: Positive-instance driven dynamic programming for treewidth. In: ESA, pp. 68:1–68:13 (2017)

    Google Scholar 

Download references

Acknowledgements

The authors would like to thank Jan Arne Telle and Fedor Fomin for helpful discussions about the topic and its presentation.

Author information

Authors and Affiliations

  1. Institute for Theoretical Computer Science, Universität zu Lübeck, Lübeck, Germany

    Max Bannach

  2. Department of Computer Science, Kiel University, Kiel, Germany

    Sebastian Berndt

Authors
  1. Max Bannach
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Sebastian Berndt
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Max Bannach .

Editor information

Editors and Affiliations

  1. Department of Computing Science, University of Alberta, Edmonton, AB, Canada

    Zachary Friggstad

  2. School of Computer Science, Carleton University, Manotick, ON, Canada

    Jörg-Rüdiger Sack

  3. Department of Computing Science, University of Alberta, Edmonton, AB, Canada

    Mohammad R Salavatipour

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Bannach, M., Berndt, S. (2019). Positive-Instance Driven Dynamic Programming for Graph Searching. In: Friggstad, Z., Sack, JR., Salavatipour, M. (eds) Algorithms and Data Structures. WADS 2019. Lecture Notes in Computer Science(), vol 11646. Springer, Cham. https://doi.org/10.1007/978-3-030-24766-9_4

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-24766-9_4

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-24765-2

  • Online ISBN: 978-3-030-24766-9

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation