Skip to main content
SpringerLink
Log in

On the Structure of Solution-Graphs for Boolean Formulas

  • Conference paper
  • First Online:
Fundamentals of Computation Theory (FCT 2015)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9210))

Included in the following conference series:

Abstract

In this work we extend the study of solution graphs and prove that for boolean formulas in a class called CPSS, all connected components are partial cubes of small dimension, a statement which was proved only for some cases in [16]. In contrast, we show that general Schaefer formulas are powerful enough to encode graphs of exponential isometric dimension and graphs which are not even partial cubes.

Our techniques shed light on the detailed structure of st-connectivity for Schaefer and connectivity for CPSS formulas, problems which were already known to be solvable in polynomial time. We refine this classification and show that the problems in these cases are equivalent to the satisfiability problem of related formulas by giving mutual reductions between (st-)connectivity and satisfiability. An immediate consequence is that st-connectivity in (undirected) solution graphs of Horn-formulas is P-complete while for 2SAT formulas st-connectivity is NL-complete.

P. Scharpfenecker—Supported by DFG grant TO 200/3-1.

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

Notes

  1. 1.

    This was proved for bijunctive formulas in [16], we prove the remaining cases of Horn and dual-Horn formulas.

  2. 2.

    Identifying two variables corresponds to replacing one of them with the other variable.

  3. 3.

    Note that as S is finite, every constraint has finite arity and therefore a solution graph of constant size.

  4. 4.

    This algorithm starts with a directed hypergraph and an initially marked set of nodes. If there is a hyperedge such that all source-nodes are marked but not all target nodes, we mark all target nodes. The algorithm finishes if there is no hyperedge which would mark a new node.

References

  1. Alon, N., Pudlák, P.: Equilateral sets in \(l_p^n\). Geom. Funct. Anal. 13(3), 467–482 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  2. Arora, S., Barak, B.: Computational Complexity: A Modern Approach, 1st edn. Cambridge University Press, New York (2009)

    Book  Google Scholar 

  3. Balcázar, J.L., Lozano, A., Torán, J.: The complexity of algorithmic problems on succinct instances. In: Baeza-Yates, R., Manber, U. (eds.) Computer Science, Research and Applications. Springer, New York (1992)

    Google Scholar 

  4. Bonsma, P., Mouawad, A.E., Nishimura, N., Raman, V.: The complexity of bounded length graph recoloring and CSP reconfiguration. In: Cygan, M., Heggernes, P. (eds.) IPEC 2014. LNCS, vol. 8894, pp. 110–121. Springer, Heidelberg (2014)

    Google Scholar 

  5. Das, B., Scharpfenecker, P., Torán, J.: Succinct encodings of graph isomorphism. In: Dediu, A.-H., Martín-Vide, C., Sierra-Rodríguez, J.-L., Truthe, B. (eds.) LATA 2014. LNCS, vol. 8370, pp. 285–296. Springer, Heidelberg (2014)

    Chapter  Google Scholar 

  6. Ekin, O., Hammer, P.L., Kogan, A.: On connected Boolean functions. Discrete Appl. Math. 96–97, 337–362 (1999)

    Article  MathSciNet  Google Scholar 

  7. Goldschlager, L.M.: The monotone and planar circuit value problems are log space complete for p. SIGACT News 9(2), 25–29 (1977)

    Article  Google Scholar 

  8. Gopalan, P., Kolaitis, P.G., Maneva, E., Papadimitriou, C.H.: The connectivity of boolean satisfiability: computational and structural dichotomies. SIAM J. Comput. 38(6), 2330–2355 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  9. Juban, L.: Dichotomy theorem for the generalized unique satisfiability problem. In: Ciobanu, G., Păun, G. (eds.) FCT 1999. LNCS, vol. 1684, pp. 327–337. Springer, Heidelberg (1999)

    Chapter  Google Scholar 

  10. Maneva, E., Mossel, E., Wainwright, M.J.: A new look at survey propagation and its generalizations. J. ACM 54(4), 1–41 (2007)

    Article  MathSciNet  Google Scholar 

  11. Mézard, M., Zecchina, R.: Random \(k\)-satisfiability problem: From an analytic solution to an efficient algorithm. Phys. Rev. E 66, 056126 (2002)

    Article  Google Scholar 

  12. Mézard, M., Ricci-Tersenghi, F., Zecchina, R.: Two solutions to diluted p-spin models and xorsat problems. J. Stat. Phys. 111(3–4), 505–533 (2003)

    Article  MATH  Google Scholar 

  13. Ovchinnikov, S.: Graphs and Cubes. Universitext. Springer, New York (2011)

    Book  MATH  Google Scholar 

  14. Papadimitriou, C.H., Yannakakis, M.: A note on succinct representations of graphs. Inf. Control 71(3), 181–185 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  15. Schaefer, T.J.: The complexity of satisfiability problems. In: Proceedings of the Tenth Annual ACM Symposium on Theory of Computing - STOC 1978, pp. 216–226. ACM Press, New York, May 1978

    Google Scholar 

  16. Schwerdtfeger, K.W.: A Computational Trichotomy for Connectivity of Boolean Satisfiability, p. 24, December 2013. http://arxiv.org/abs/1312.4524

  17. Veith, H.: Languages represented by Boolean formulas. Inf. Process. Lett. 63(5), 251–256 (1997)

    Article  MathSciNet  Google Scholar 

  18. Veith, H.: How to encode a logical structure by an OBDD. In: Proceedings of the 13th IEEE Conference on Computational Complexity, pp. 122–131. IEEE Computer Society (1998)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Theoretical Computer Science, University of Ulm, Ulm, Germany

    Patrick Scharpfenecker

Authors
  1. Patrick Scharpfenecker
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Patrick Scharpfenecker .

Editor information

Editors and Affiliations

  1. Université Paris Diderot, Paris, France

    Adrian Kosowski

  2. Université Bordeaux 1, Talence, France

    Igor Walukiewicz

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer International Publishing Switzerland

About this paper

Cite this paper

Scharpfenecker, P. (2015). On the Structure of Solution-Graphs for Boolean Formulas. In: Kosowski, A., Walukiewicz, I. (eds) Fundamentals of Computation Theory. FCT 2015. Lecture Notes in Computer Science(), vol 9210. Springer, Cham. https://doi.org/10.1007/978-3-319-22177-9_10

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-22177-9_10

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-22176-2

  • Online ISBN: 978-3-319-22177-9

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation