Blowing Holes in Various Aspects of Computational Problems, with Applications to Constraint Satisfaction

  • Peter Jonsson
  • Victor Lagerkvist
  • Gustav Nordh
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8124)


We consider methods for constructing NP-intermediate problems under the assumption that P ≠ NP. We generalize Ladner’s original method for obtaining NP-intermediate problems by using parameters with various characteristics. In particular, this generalization allows us to obtain new insights concerning the complexity of CSP problems. We begin by fully characterizing the problems that admit NP-intermediate subproblems for a broad and natural class of parameterizations, and extend the result further such that structural CSP restrictions based on parameters that are hard to compute (such as tree-width) are covered. Hereby we generalize a result by Grohe on width parameters and NP-intermediate problems. For studying certain classes of problems, including CSPs parameterized by constraint languages, we consider more powerful parameterizations. First, we identify a new method for obtaining constraint languages Γ such that CSP(Γ) are NP-intermediate. The sets Γ can have very different properties compared to previous constructions (by, for instance, Bodirsky & Grohe) and provides insights into the algebraic approach for studying the complexity of infinite-domain CSPs. Second, we prove that the propositional abduction problem parameterized by constraint languages admits NP-intermediate problems. This settles an open question posed by Nordh & Zanuttini.


Polynomial Time Turing Machine Constraint Satisfaction Constraint Satisfaction Problem Computational Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Arora, S., Barak, B.: Computational Complexity: A Modern Approach, 1st edn. Cambridge University Press, New York (2009)CrossRefGoogle Scholar
  2. 2.
    Arnborg, S., Corneil, D., Proskurowski, A.: Complexity of finding embeddings in a k-tree. SIAM Journal on Matrix Analysis and Applications 8(2), 277–284 (1987)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Bodirsky, M.: Complexity Classification in Infinite-Domain Constraint Satisfaction. Habilitation thesis. Univ. Paris 7 (2012)Google Scholar
  4. 4.
    Bodirsky, M., Grohe, M.: Non-dichotomies in constraint satisfaction complexity. In: Aceto, L., Damgård, I., Goldberg, L.A., Halldórsson, M.M., Ingólfsdóttir, A., Walukiewicz, I. (eds.) ICALP 2008, Part II. LNCS, vol. 5126, pp. 184–196. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  5. 5.
    Bodlaender, H.: A linear-time algorithm for finding tree-decompositions of small treewidth. SIAM Journal on Computing 25(6), 1305–1317 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bulatov, A., Jeavons, P., Krokhin, A.: Classifying the computational complexity of constraints using finite algebras. SIAM Journal on Computing 34(3), 720–742 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Chen, J., Chor, B., Fellows, M., Huang, X., Juedes, D., Kanj, I., Xia, G.: Tight lower bounds for certain parameterized np-hard problems. In: Proc. IEEE Conference on Computational Complexity (CCC 2004), pp. 150–160 (2004)Google Scholar
  8. 8.
    Chen, J., Huang, X., Kanj, I., Xia, G.: Linear fpt reductions and computational lower bounds. In: Proc. 36th ACM Symposium on Theory of Computing (STOC-2004), pp. 212–221 (2004)Google Scholar
  9. 9.
    Chen, Y., Thurley, M., Weyer, M.: Understanding the complexity of induced subgraph isomorphisms. In: Aceto, L., Damgård, I., Goldberg, L.A., Halldórsson, M.M., Ingólfsdóttir, A., Walukiewicz, I. (eds.) ICALP 2008, Part I. LNCS, vol. 5125, pp. 587–596. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  10. 10.
    Creignou, N., Schmidt, J., Thomas, M.: Complexity of propositional abduction for restricted sets of boolean functions. In: Proc. 12th International Conference on the Principles of Knowledge Representation and Reasoning, KR 2010 (2010)Google Scholar
  11. 11.
    Dechter, R.: Constraint Processing. Elsevier Morgan Kaufmann (2003)Google Scholar
  12. 12.
    Dell, H., van Melkebeek, D.: Satisfiability allows no nontrivial sparsification unless the polynomial-time hierarchy collapses. In: Proc. 42nd ACM Symposium on Theory of Computing (STOC 2010), pp. 251–260 (2010)Google Scholar
  13. 13.
    Eiter, T., Gottlob, G.: The complexity of logic-based abduction. Journal of the ACM 42(1), 3–42 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Garey, M., Johnson, D.: “Strong” NP-completeness results: motivation, examples and implications. Journal of the ACM 25(3), 499–508 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Grohe, M.: The complexity of homomorphism and constraint satisfaction problems seen from the other side. Journal of the ACM 54(1), article 1 (2007)Google Scholar
  16. 16.
    Grohe, M., Marx, D.: Constraint solving via fractional edge covers. In: Proc. 17th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA-2006), pp. 289–298 (2006)Google Scholar
  17. 17.
    Jonsson, P., Lagerkvist, V., Nordh, G., Zanuttini, B.: Complexity of SAT problems, clone theory and the exponential time hypothesis. In: Proc. the Twenty-Fourth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2013 (2013)Google Scholar
  18. 18.
    Jonsson, P., Lööw, T.: Computational complexity of linear constraints over the integers. Artificial Intelligence 195, 44–62 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Karp, R.M.: Reducibility Among Combinatorial Problems. In: Miller, R.E., Thatcher, J.W. (eds.) Complexity of Computer Computations, pp. 85–103. Plenum Press (1972)Google Scholar
  20. 20.
    Ladner, R.: On the structure of polynomial time reducibility. Journal of the ACM 22, 155–171 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Marx, D.: Approximating fractional hypertree width. ACM Transactions on Algorithms 6(2) (2010)Google Scholar
  22. 22.
    Marx, D.: Tractable hypergraph properties for constraint satisfaction and conjunctive queries. In: Proc. 42nd ACM Symposium on Theory of Computing (STOC 2010), pp. 735–744 (2010)Google Scholar
  23. 23.
    Nordh, G., Zanuttini, B.: What makes propositional abduction tractable. Artificial Intelligence 172, 1245–1284 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Papadimitriou, C.H.: Computational Complexity. Addison-Wesley (1994)Google Scholar
  25. 25.
    Schöning, U.: A uniform approach to obtain diagonal sets in complexity classes. Theoretical Computer Science 18, 95–103 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Lau, D.: Function Algebras on Finite Sets: Basic Course on Many-Valued Logic and Clone Theory. Springer Monographs in Mathematics. Springer-Verlag New York, Inc., Secaucus (2006)Google Scholar

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© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Peter Jonsson
    • 1
  • Victor Lagerkvist
    • 1
  • Gustav Nordh
    • 1
  1. 1.Department of Computer and Information ScienceLinköping UniversitySweden

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