Kernelization of Constraint Satisfaction Problems: A Study Through Universal Algebra

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10416)

Abstract

A kernelization algorithm for a computational problem is a procedure which compresses an instance into an equivalent instance whose size is bounded with respect to a complexity parameter. For the constraint satisfaction problem (CSP), there exist many results concerning upper and lower bounds for kernelizability of specific problems, but it is safe to say that we lack general methods to determine whether a given problem admits a kernel of a particular size. In this paper, we take an algebraic approach to the problem of characterizing the kernelization limits of NP-hard CSP problems, parameterized by the number of variables. Our main focus is on problems admitting linear kernels, as has, somewhat surprisingly, previously been shown to exist. We show that a finite-domain CSP problem has a kernel with O(n) constraints if it can be embedded (via a domain extension) into a CSP which is preserved by a Maltsev operation. This result utilise a variant of the simple algorithm for Maltsev constraints. In the complementary direction, we give indication that the Maltsev condition might be a complete characterization for Boolean CSPs with linear kernels, by showing that an algebraic condition that is shared by all problems with a Maltsev embedding is also necessary for the existence of a linear kernel unless NP \(\subseteq \) co-NP/poly.

References

  1. 1.
    Bulatov, A.: A dichotomy theorem for nonuniform CSPs. CoRR, abs/1703.03021 (2017)Google Scholar
  2. 2.
    Bulatov, A., Dalmau, V.: A simple algorithm for Mal’tsev constraints. SICOMP 36(1), 16–27 (2006)CrossRefMATHGoogle Scholar
  3. 3.
    Bulatov, A., Jeavons, P., Krokhin, A.: Classifying the complexity of constraints using finite algebras. SICOMP 34(3), 720–742 (2005)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Dalmau, V., Jeavons, P.: Learnability of quantified formulas. TCS 306(1–3), 485–511 (2003)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Dell, H., van Melkebeek, D.: Satisfiability allows no nontrivial sparsification unless the polynomial-time hierarchy collapses. J. ACM 61(4), 23:1–23:27 (2014)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Dyer, M., Richerby, D.: An effective dichotomy for the counting constraint satisfaction problem. SICOMP 42(3), 1245–1274 (2013)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Feder, T., Vardi, M.: The computational structure of monotone monadic SNP and constraint satisfaction: a study through datalog and group theory. SICOMP 28(1), 57–104 (1998)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Geiger, D.: Closed systems of functions and predicates. Pac. J. Math. 27(1), 95–100 (1968)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Goldstern, M., Pinsker, M.: A survey of clones on infinite sets. Algebra Univers. 59(3), 365–403 (2008)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Impagliazzo, R., Paturi, R., Zane, F.: Which problems have strongly exponential complexity? J. Comput. Syst. Sci. 63, 512–530 (2001)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Jansen, B.M.P., Pieterse, A.: Sparsification upper and lower bounds for graphs problems and not-all-equal SAT. In: Proceedings of IPEC 2015, Patras, Greece (2015)Google Scholar
  12. 12.
    Jansen, B.M.P., Pieterse, A.: Optimal sparsification for some binary CSPs using low-degree polynomials. In: Proceedings of MFCS 2016, vol. 58, pp. 71:1–71:14 (2016)Google Scholar
  13. 13.
    Jeavons, P.: On the algebraic structure of combinatorial problems. TCS 200, 185–204 (1998)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Jeavons, P., Cohen, D., Gyssens, M.: A unifying framework for tractable constraints. In: Montanari, U., Rossi, F. (eds.) CP 1995. LNCS, vol. 976, pp. 276–291. Springer, Heidelberg (1995). doi:10.1007/3-540-60299-2_17 Google Scholar
  15. 15.
    Jeavons, P., Cohen, D., Gyssens, M.: Closure properties of constraints. JACM 44(4), 527–548 (1997)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Jonsson, P., Lagerkvist, V., Nordh, G., Zanuttini, B.: Strong partial clones and the time complexity of SAT problems. JCSS 84, 52–78 (2017)MathSciNetMATHGoogle Scholar
  17. 17.
    Kratsch, S., Marx, D., Wahlström, M.: Parameterized complexity and kernelizability of max ones and exact ones problems. TOCT 8(1), 1 (2016)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Kratsch, S., Wahlström, M.: Preprocessing of min ones problems: a dichotomy. In: Abramsky, S., Gavoille, C., Kirchner, C., Meyer auf der Heide, F., Spirakis, P.G. (eds.) ICALP 2010. LNCS, vol. 6198, pp. 653–665. Springer, Heidelberg (2010). doi:10.1007/978-3-642-14165-2_55 CrossRefGoogle Scholar
  19. 19.
    Krokhin, A.A., Marx, D.: On the hardness of losing weight. ACM Trans. Algorithms 8(2), 19 (2012)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Lagerkvist, V., Wahlström, M.: The power of primitive positive definitions with polynomially many variables. JLC 27, 1465–1488 (2016)Google Scholar
  21. 21.
    Lagerkvist, V., Wahlström, M.: Kernelization of constraint satisfaction problems: a study through universal algebra. ArXiv e-prints, June 2017Google Scholar
  22. 22.
    Lagerkvist, V., Wahlström, M., Zanuttini, B.: Bounded bases of strong partial clones. In: Proceedings of ISMVL 2015 (2015)Google Scholar
  23. 23.
    Lau, D.: Function Algebras on Finite Sets: Basic Course on Many-Valued Logic and Clone Theory (Springer Monographs in Mathematics). Springer, New York (2006)MATHGoogle Scholar
  24. 24.
    Marx, D.: Parameterized complexity of constraint satisfaction problems. Comput. Complex. 14(2), 153–183 (2005)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Nemhauser, G.L., Trotter, L.E.: Vertex packings: structural properties and algorithms. Math. Program. 8(1), 232–248 (1975)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Rafiey, A., Kinne, J., Feder, T.: Dichotomy for digraph homomorphism problems. CoRR, abs/1701.02409 (2017)Google Scholar
  27. 27.
    Zhuk, D.: The proof of CSP dichotomy conjecture. CoRR, abs/1704.01914 (2017)Google Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Institut für AlgebraTU DresdenDresdenGermany
  2. 2.Department of Computer ScienceRoyal Holloway, University of LondonEghamGreat Britain

Personalised recommendations