Advertisement

Solution-Graphs of Boolean Formulas and Isomorphism

  • Patrick ScharpfeneckerEmail author
  • Jacobo Torán
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9710)

Abstract

The solution graph of a Boolean formula on n variables is the subgraph of the hypercube \(H_n\) induced by the satisfying assignments of the formula. The structure of solution graphs has been the object of much research in recent years since it is important for the performance of SAT-solving procedures based on local search. Several authors have studied connectivity problems in such graphs focusing on how the structure of the original formula might affect the complexity of the connectivity problems in the solution graph.

In this paper we study the complexity of the isomorphism problem of solution graphs of Boolean formulas and we investigate how this complexity depends on the formula type.

We observe that for general formulas the solution graph isomorphism problem can be solved in exponential time while in the cases of 2CNF formulas, as well as for CPSS formulas, the problem is in the counting complexity class \(\text {C}_=\text {P} \), a subclass of PSPACE. We also prove a strong property on the structure of solution graphs of Horn formulas showing that they are just unions of partial cubes.

In addition we give a \(\text {PSPACE} \) lower bound for the problem on general Boolean functions. We prove that for 2CNF, as well as for CPSS formulas the solution graph isomorphism problem is hard for \(\text {C}_=\text {P} \) under polynomial time many one reductions, thus matching the given upper bound.

Keywords

Solution graph Isomorphism Counting Partial cube 

References

  1. 1.
    Achlioptas, D., Coja-Oghlan, A., Ricci-Tersenghi, F.: On the solution-space geometry of random constraint satisfaction problems. Random Struct. Algorithms 38(3), 251–268 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Agrawal, M., Thierauf, T.: The Boolean isomorphism problem. In: Proceedings of 37th Conference on Foundations of Computer Science, pp. 422–430. IEEE Computer Society Press (1996)Google Scholar
  3. 3.
    Babai, L.: Graph isomorphism in quasipolynomial time. In: Proceedings of 48th Annual Symposium on the Theory of Computing, STOC (2016)Google Scholar
  4. 4.
    Bandelt, H.-J., van de Vel, M.: Embedding topological median algebras in products of dendrons. Proc. London Math. Soc. 3(58), 439–453 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bandelt, H.J., Chepoi, V.: Metric graph theory and geometry: a survey. Contemp. Math. 453, 49–86 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Böhler, E., Hemaspaandra, E., Reith, S., Vollmer, H.: Equivalence and isomorphism for boolean constraint satisfaction. In: Bradfield, J.C. (ed.) CSL 2002 and EACSL 2002. LNCS, vol. 2471, pp. 412–426. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  7. 7.
    Curticapean, R.: Parity separation: a scientifically proven method for permanent weight loss. arXiv preprint. arXiv:1511.07480 (2015)
  8. 8.
    Gableske, O.: SAT Solving with Message Passing. Ph.D. thesis, University of Ulm (2016)Google Scholar
  9. 9.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Green, F.: On the power of deterministic reductions to C=P. Math. Syst. Theor. 26(2), 215–233 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Jenner, B., Köbler, J., McKenzie, P., Torán, J.: Completeness results for graph isomorphism. J. Comput. Syst. Sci. 66(3), 549–566 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Köbler, J., Schöning, U., Torán, J.: Graph isomorphism is low for PP. Comput. Complex. 2(4), 301–330 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Köbler, J., Schöning, U., Torán, J.: The Graph Isomorphism Problem: its Structural Complexity. Birkhauser, Boston (1993)CrossRefzbMATHGoogle Scholar
  14. 14.
    Luks, E.M.: Isomorphism of graphs of bounded valence can be tested in polynomial time. J. Comput. Syst. Sci. 25(1), 42–65 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Mézard, M., Mora, T., Zecchina, R.: Clustering of solutions in the random satisfiability problem. Phys. Rev. Lett. 94(19), 197205 (2005)CrossRefGoogle Scholar
  16. 16.
    Ovchinnikov, S.: Graphs and Cubes. Universitext. Springer, New York (2011)CrossRefzbMATHGoogle Scholar
  17. 17.
    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 (1978)Google Scholar
  18. 18.
    Scharpfenecker, P.: On the structure of solution-graphs for boolean formulas. In: Kosowski, A., Walukiewicz, I. (eds.) FCT 2015. LNCS, vol. 9210, pp. 118–130. Springer, Heidelberg (2015)CrossRefGoogle Scholar
  19. 19.
    Schwerdtfeger, K.W.: A computational trichotomy for connectivity of boolean satisfiability. JSAT 8(3/4), 173–195 (2014)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Valiant, L.G.: The complexity of computing the permanent. Theor. Comput. Sci. 8(2), 189–201 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Valiant, L.G.: The complexity of enumeration and reliability problems. SIAM J. Comput. 8, 410–421 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Wagner, K.W.: The complexity of combinatorial problems with succinct input representation. Acta Informatica 23(3), 325–356 (1986)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Institute of Theoretical Computer ScienceUniversity of UlmUlmGermany

Personalised recommendations