All or Nothing: Toward a Promise Problem Dichotomy for Constraint Problems

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

Abstract

We show that intractability of the constraint satisfaction problem over a fixed finite constraint language can, in all known cases, be replaced by an infinite hierarchy of intractable promise problems of increasingly disparate promise conditions. The instances are guaranteed to either have no solutions at all, or to be k-robustly satisfiable (for any fixed k), meaning that every “reasonable” partial instantiation on k variables extends to a solution.

Keywords

Constraint satisfaction problem Dichotomy Robust satisfiability Promise problem Quasivariety Universal horn class 

References

  1. 1.
    Allender, E., Bauland, M., Immerman, N., Schnoor, H., Vollmer, H.: The complexity of satisfiability problems: refining Schaefer’s Theorem. J. Comput. System Sci. 75, 245–254 (2009)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Almeida, J.: Finite Semigroups and Universal Algebra. World Scientific, Singapore (1994)MATHGoogle Scholar
  3. 3.
    Abramsky, S., Gottlob, G., Kolaitis, P.G.: Robust constraint satisfaction and local hidden variables in quantum mechanics. In: IJCAI 2013, pp. 440–446 (2013)Google Scholar
  4. 4.
    Arora, S., Safra, S.: Probabilistic checking of proofs: a new characterization of NP. J. ACM 45(1), 70–122 (1998)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Barto, L., Kozik, M.: Constraint satisfaction problems of bounded width. In: Proceedings of FOCS 2009 (2009)Google Scholar
  6. 6.
    Barto, L., Kozik, M., Niven, T.: The CSP dichotomy holds for digraphs with no sources and no sinks (a positive answer to a conjecture of Bang-Jensen and Hell), SIAM J. Comput. 38, 1782–1802 (2008/2009)Google Scholar
  7. 7.
    Beacham, A., Culberson, J.: On the complexity of unfrozen problems. Disc. Appl. Math. 153, 3–24 (2005)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Bulatov, A.A.: A dichotomy theorem for constraint satisfaction problems on a 3-element set. J. ACM 53, 66–120 (2006)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Bulatov, A.A.: Complexity of conservative constraint satisfaction problems. ACM Trans. Comput. Log. 12(4), 24 (2011)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Bulatov, A.A.: A dichotomy theorem for nonuniform CSPs. arXiv:1703.03021v2
  11. 11.
    Bulatov, A.A., Jeavons, P.G., Krokhin, A.: Classifying the complexity of constraints using finite algebras. SIAM J. Comput. 34(3), 720–742 (2005)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Caicedo, X.: Finitely axiomatizable quasivarieties of graphs. Algebra Univers. 34, 314–321 (1995)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Chen, H., Larose, B.: Asking the metaquestions in constraint tractability. arxiv:1604.00932
  14. 14.
    Feder, T., Vardi, M.Y.: The computational structure of monotone monadic SNP and constraint satisfaction: a study through datalog and group theory. SIAM J. Comput. 28(1), 57–104 (1998)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W.H. Freeman & Co., New York (1979)MATHGoogle Scholar
  16. 16.
    Geiger, D.: Closed systems of functions and predicates. Pacific J. Math. 27, 95–100 (1968)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Gorbunov, V.A.: Algebraic Theory of Quasivarieties. Consultants Bureau, New York (1998)MATHGoogle Scholar
  18. 18.
    Gottlob, G.: On minimal constraint networks. In: Lee, J. (ed.) CP 2011. LNCS, vol. 6876, pp. 325–339. Springer, Heidelberg (2011). doi:10.1007/978-3-642-23786-7_26 CrossRefGoogle Scholar
  19. 19.
    Goldreich, O.: On promise problems: a survey. In: Goldreich, O., Rosenberg, A.L., Selman, A.L. (eds.) Theoretical Computer Science. LNCS, vol. 3895, pp. 254–290. Springer, Heidelberg (2006). doi:10.1007/11685654_12 CrossRefGoogle Scholar
  20. 20.
    Ham, L.: A gap trichotomy theorem for Boolean constraint problems: extending Schaefer’s theorem. In: ISAAC 2016, pp. 36: 1–36: 12 (2016)Google Scholar
  21. 21.
    Ham, L.: Gap theorems for robust satisfiability of constraint problems: Boolean CSPs and beyond. Theoret. Comp. Sci. 676, 69–91 (2017)CrossRefMATHGoogle Scholar
  22. 22.
    Hell, P., Nešetřil, J.: On the complexity of H-colouring. J. Combin. Theory Ser. B 48(1), 92–110 (1990)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Idziak, P., Markovic, P., McKenzie, R., Valeriote, M., Willard, R.: Tractability and learnability arising from algebras with few subpowers. SIAM J. Comput. 39, 3023–3037 (2010)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Jackson, M.: Flexible constraint satisfiability and a problem in semigroup theory. arXiv:1512.03127
  25. 25.
    Jackson, M., Kowalski, T., Niven, T.: Digraph related constructions and the complexity of digraph homomorphism problems. Int. J. Algebra Comput. 26, 1395–1433 (2016)CrossRefMATHGoogle Scholar
  26. 26.
    Jackson, M., Trotta, B.: Constraint satisfaction, irredundant axiomatisability and continuous colouring. Stud. Logica. 101, 65–94 (2013)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Jeavons, P.: On the algebraic structure of combinatorial problems. Theor. Comput. Sci. 200(1–2), 185–204 (1998)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Jeavons, P., Cohen, D.A., Martin, C.: Cooper.: constraints, consistency and closure. Artif. Intell. 101(1–2), 251–265 (1998)CrossRefMATHGoogle Scholar
  29. 29.
    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
  30. 30.
    Jeavons, P., Cohen, D.A., Gyssens, M.: Closure properties of constraints. J. ACM 44, 527–548 (1997)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Jeavons, P., Cohen, D.A., Pearson, J.: Constraints and universal algebra. Ann. Math. Artif. Intell. 24(1–4), 51–67 (1998)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Jonsson, P., Krokhin, A.: Recognizing frozen variables in constraint satisfaction problems. Theoret. Comp. Sci. 329, 93–113 (2004)MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Larose, B., Tesson, P.: Universal algebra and hardness results for constraint satisfaction problems. Theoret. Comput. Sci. 410, 1629–1647 (2009)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Maltsev, A.I.: Algebraic Systems. Springer, Heidelberg (1973)Google Scholar
  35. 35.
    Maróti, M., McKenzie, R.: Existence theorems for weakly symmetric operations. Algebra Univers. 59, 463–489 (2008)MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Marques-Silva, J.P., Sakallah, K.A.: GRASP: a search algorithm for propositional satisfiability. IEEE Trans. Comput. 48, 506–521 (1999)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Monasson, R., Zecchina, R., Kirkpatrick, S., Selman, B., Troyansky, L.: Determining computational complexity from characteristic phase transitions. Nature 400, 133–137 (1998)MathSciNetMATHGoogle Scholar
  38. 38.
    Nešetřil, J., Pultr, A.: On classes of relations and graphs determined by subobjects and factorobjects. Disc. Math. 22, 287–300 (1978)MathSciNetCrossRefMATHGoogle Scholar
  39. 39.
    Rafiey, A., Kinne, J., Feder, T.: Dichotomy for digraph homomorphism problems, arXiv:1701.02409v2
  40. 40.
    Schaefer, T.J.: The complexity of satisfiability problems. In: STOC , pp. 216–226 (1978)Google Scholar
  41. 41.
    Valeriote, M.A.: A subalgebra intersection property for congruence distributive varieties. Canad. J. Math. 61, 451–464 (2009)MathSciNetCrossRefMATHGoogle Scholar
  42. 42.
    Zhuk, D.: The proof of the CSP dichotomy conjecture, arXiv:1704.01914

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsLa Trobe UniversityMelbourneAustralia

Personalised recommendations