Skip to main content

Reducing Treewidth for SAT-Related Problems Using Simple Liftings

  • Conference paper
  • First Online:
Combinatorial Optimization (ISCO 2024)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14594))

Included in the following conference series:

  • 357 Accesses

Abstract

Tree decompositions are a powerful tool to obtain parameterized algorithms, in particular to solve different variants of the satisfiability problem. Most algorithms are based on a tree decomposition of the so called primal graph. Variants of the satisfiability problem that allow parameterized algorithms in the treewidth of the primal graph are for example Model Counting, MaxSat or QBF.

To obtain efficient algorithms in practice, reducing the size of the instance by preprocessing is a very important technique and hence is highly investigated. In this paper, we investigate how preprocessing techniques can be used to reduce the parameter of a parameterized algorithm other than the size of the instance. In particular, we look at satisfiability and related problems and try to preprocess the formula in order to reduce the treewidth of the resulting primal graph. To the best of our knowledge, this is the first such approach.

We show how to compute a set of auxiliary variables and an equisatisfiable (w.r.t. the original variables) formula using those such that the treewidth of the resulting primal graph is minimal under all sets of auxiliary variables. To reach this goal, we restrict our attention to auxiliary variables such that their value has to be the value of a subclause of the formula for each satisfying truth assignment.

We implemented our approach and evaluated it on standard benchmark instances. While our approach is able to reduce the treewidth of around \(10\%\) of the instances, there is no clear improvement in the running time when solving the formula, due to the dependence of the practical efficiency of the solver on the structure of the formula.

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

Access this chapter

Subscribe and save

Springer+ Basic
$34.99 /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
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

References

  1. Biere, A., Järvisalo, M., Kiesl, B.: Preprocessing in SAT solving. In: Biere, A., Heule, M., van Maaren, H., Walsh, T. (eds.) Handbook of Satisfiability, 2nd edn., vol. 336 of Frontiers in Artificial Intelligence and Applications, pp. 391–435. IOS Press (2021)

    Google Scholar 

  2. Bodlaender, H.L., Koster, A.M.: Treewidth computations I. Upper bounds. Inf. Comput. 208(3), 259–275 (2010)

    Article  MathSciNet  Google Scholar 

  3. Charwat, G., Woltran, S.: Expansion-based QBF solving on tree decompositions. Fundam. Informaticae 167(1–2), 59–92 (2019)

    Article  MathSciNet  Google Scholar 

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

    Book  Google Scholar 

  5. Fichte, J.K., Hecher, M., Morak, M., Thier, P., Woltran, S.: Solving projected model counting by utilizing treewidth and its limits. Artif. Intell. 314, 103810 (2023)

    Article  MathSciNet  Google Scholar 

  6. Fichte, J.K., Hecher, M., Pfandler, A.: Lower bounds for qbfs of bounded treewidth. In: Hermanns, H., Zhang, L., Kobayashi, N., Miller, D. (eds.) LICS 2020: 35th Annual ACM/IEEE Symposium on Logic in Computer Science, Saarbrücken, Germany, 8–11 July 2020, pp. 410–424. ACM (2020)

    Google Scholar 

  7. Habet, D., Paris, L., Terrioux, C.: A tree decomposition based approach to solve structured SAT instances. In: ICTAI 2009, 21st IEEE International Conference on Tools with Artificial Intelligence, Newark, New Jersey, USA, 2–4 November 2009, pp. 115–122. IEEE Computer Society (2009)

    Google Scholar 

  8. Hecher, M.: Treewidth-aware reductions of normal ASP to SAT - is normal ASP harder than SAT after all? Artif. Intell. 304, 103651 (2022)

    Article  MathSciNet  Google Scholar 

  9. Hoder, K., Voronkov, A.: The 481 ways to split a clause and deal with propositional variables. In: Bonacina, M.P. (ed.) Automated Deduction - CADE-24 - 24th International Conference on Automated Deduction, Lake Placid, NY, USA, 9–14 June 2013. Proceedings, vol. 7898 of Lecture Notes in Computer Science, pp. 450–464. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-38574-2_33

  10. Ihalainen, H., Berg, J., Järvisalo, M.: Clause redundancy and preprocessing in maximum satisfiability. In: Blanchette, J., Kovács, L., Pattinson, D. (eds.) Automated Reasoning - 11th International Joint Conference, IJCAR 2022, Haifa, Israel, 8–10 August 2022, Proceedings, vol. 13385 of Lecture Notes in Computer Science, pp. 75–94. Springer, Heidelberg (2022). https://doi.org/10.1007/978-3-031-10769-6_6

  11. Korhonen, T., Järvisalo, M.: Integrating tree decompositions into decision heuristics of propositional model counters (short paper). In: Michel, L.D. (ed.) 27th International Conference on Principles and Practice of Constraint Programming, CP 2021, Montpellier, France (Virtual Conference), 25–29 October 2021, vol. 210 of LIPIcs, pp. 8:1–8:11. Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)

    Google Scholar 

  12. Lagniez, J., Marquis, P.: On preprocessing techniques and their impact on propositional model counting. J. Autom. Reason. 58(4), 413–481 (2017)

    Article  MathSciNet  Google Scholar 

  13. Lokshtanov, D., Panolan, F., Ramanujan, M.S.: Backdoor sets on nowhere dense SAT. In: Bojańczyk, M., Merelli, E., Woodruff, D.P. (eds.) 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022), vol. 229 of Leibniz International Proceedings in Informatics (LIPIcs), Dagstuhl, Germany, pp. 91:1–91:20. Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

    Google Scholar 

  14. Lonsing, F.: Qbfrelay, qratpre+, and depqbf: incremental preprocessing meets search-based QBF solving. J. Satisf. Boolean Model. Comput. 11(1), 211–220 (2019)

    MathSciNet  Google Scholar 

  15. Lonsing, F., Egly, U.: Qratpre+: effective QBF preprocessing via strong redundancy properties. In: Janota, M., Lynce, I. (eds.) Theory and Applications of Satisfiability Testing - SAT 2019 - 22nd International Conference, SAT 2019, Lisbon, Portugal, July 9-12, 2019, Proceedings, vol. 11628 of Lecture Notes in Computer Science, pp. 203–210. Springer, Heidelberg (2019). https://doi.org/10.1007/978-3-030-24258-9_14

  16. Mählmann, N., Siebertz, S., Vigny, A.: Recursive backdoors for SAT. CoRR arxiv:2102.04707 (2021)

  17. Markov, I.L., Shi, Y.: Constant-degree graph expansions that preserve treewidth. Algorithmica 59(4), 461–470 (2011)

    Article  MathSciNet  Google Scholar 

  18. Sæther, S.H., Telle, J.A., Vatshelle, M.: Solving #SAT and MAXSAT by dynamic programming. J. Artif. Intell. Res. 54, 59–82 (2015)

    Article  MathSciNet  Google Scholar 

  19. Samer, M., Szeider, S.: Algorithms for propositional model counting. J. Disc. Algor. 8(1), 50–64 (2010)

    Article  MathSciNet  Google Scholar 

  20. Slivovsky, F., Szeider, S.: A faster algorithm for propositional model counting parameterized by incidence treewidth. In: Pulina, L., Seidl, M. (eds.) Theory and Applications of Satisfiability Testing - SAT 2020 - 23rd International Conference, Alghero, Italy, July 3-10, 2020, Proceedings, vol. 12178 of Lecture Notes in Computer Science, pp. 267–276. Springer, Heidelberg (2020). https://doi.org/10.1007/978-3-030-51825-7_19

  21. Soos, M., Meel, K.S.: Arjun: an efficient independent support computation technique and its applications to counting and sampling. In: ICCAD 2022, New York, NY, USA. Association for Computing Machinery (2022)

    Google Scholar 

  22. Wallon, R., Mengel, S.: Revisiting graph width measures for CNF-encodings. J. Artif. Intell. Res. 67, 409–436 (2020)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding authors

Correspondence to Ernst Althaus or Daniela Schnurbusch .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2024 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Althaus, E., Schnurbusch, D. (2024). Reducing Treewidth for SAT-Related Problems Using Simple Liftings. In: Basu, A., Mahjoub, A.R., Salazar González, J.J. (eds) Combinatorial Optimization. ISCO 2024. Lecture Notes in Computer Science, vol 14594. Springer, Cham. https://doi.org/10.1007/978-3-031-60924-4_14

Download citation

  • DOI: https://doi.org/10.1007/978-3-031-60924-4_14

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-60923-7

  • Online ISBN: 978-3-031-60924-4

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics