As Close as It Gets

  • Mike Behrisch
  • Miki Hermann
  • Stefan Mengel
  • Gernot SalzerEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9627)


We study the minimum Hamming distance between distinct satisfying assignments of a conjunctive input formula over a given set of Boolean relations (\(\mathsf {MinSolutionDistance}\), \(\mathsf {MSD}\)). We present a complete classification of the complexity of this optimization problem with respect to the relations admitted in the formula. We give polynomial time algorithms for several classes of constraint languages. For all other cases we prove hardness or completeness with respect to \(\text {poly-APX}\), or \(\mathrm {NPO}\), or equivalence to a well-known hard optimization problem.


Boolean Relation Minimum Hamming Distance Constraint Language Dual Horn Binary Clauses 
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  1. 1.
    Aspvall, B., Plass, M.R., Tarjan, R.E.: A linear-time algorithm for testing the truth of certain quantified Boolean formulas. Inf. Process. Lett. 8(3), 121–123 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Ausiello, G., Crescenzi, P., Gambosi, G., Kann, V., Marchetti-Spaccamela, A., Protasi, M.: Complexity and Approximation: Combinatorial Optimization Problems and Their Approximability Properties. Springer, New York (1999)CrossRefzbMATHGoogle Scholar
  3. 3.
    Baker, K.A., Pixley, A.F.: Polynomial interpolation and the Chinese Remainder Theorem for algebraic systems. Mathematische Zeitschrift 143(2), 165–174 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Behrisch, M., Hermann, M., Mengel, S., Salzer, G.: Minimal distance of propositional models (2015). abs/1502.06761
  5. 5.
    Behrisch, M., Hermann, M., Mengel, S., Salzer, G.: Give me another one!. In: Elbassioni, K., Makino, K. (eds.) ISAAC 2015. LNCS, vol. 9472, pp. 664–676. Springer, Heidelberg (2015). doi: 10.1007/978-3-662-48971-0_56CrossRefGoogle Scholar
  6. 6.
    Böhler, E., Creignou, N., Reith, S., Vollmer, H.: Playing with Boolean blocks, part II: constraint satisfaction problems. SIGACT News 35(1), 22–35 (2004)CrossRefGoogle Scholar
  7. 7.
    Böhler, E., Reith, S., Schnoor, H., Vollmer, H.: Bases for Boolean co-clones. Inf. Process. Lett. 96(2), 59–66 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Creignou, N., Khanna, S., Sudan, M.: Complexity Classifications of Boolean Constraint Satisfaction Problems. SIAM Monographs on Discrete Mathematics and Applications, vol. 7. SIAM, Philadelphia (2001)CrossRefzbMATHGoogle Scholar
  9. 9.
    Crescenzi, P., Rossi, G.: On the Hamming distance of constraint satisfaction problems. Theor. Comput. Sci. 288(1), 85–100 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Dumer, I., Micciancio, D., Sudan, M.: Hardness of approximating the minimum distance of a linear code. IEEE Trans. Inf. Theory 49(1), 22–37 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Jeavons, P., Cohen, D., Gyssens, M.: Closure properties of constraints. J. Assoc. Comput. Mach. 44(4), 527–548 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    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)CrossRefGoogle Scholar
  13. 13.
    Lagerkvist, V.: Weak bases of Boolean co-clones. Inf. Process. Lett. 114(9), 462–468 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Schaefer, T.J.: The complexity of satisfiability problems. In: Proceedings of the Tenth Annual ACM Symposium on Theory of Computing, STOC 1978, San Diego, California, pp. 216–226. ACM, New York (1978).
  15. 15.
    Schnoor, H., Schnoor, I.: Partial polymorphisms and constraint satisfaction problems. In: Creignou, N., Kolaitis, P.G., Vollmer, H. (eds.) Complexity of Constraints. LNCS, vol. 5250, pp. 229–254. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  16. 16.
    Warshall, S.: A theorem on Boolean matrices. J. Assoc. Comput. Mach. 9(1), 11–12 (1962)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Mike Behrisch
    • 1
  • Miki Hermann
    • 2
  • Stefan Mengel
    • 2
  • Gernot Salzer
    • 1
    Email author
  1. 1.Technische Universität WienViennaAustria
  2. 2.LIX (UMR CNRS 7161)École PolytechniquePalaiseauFrance

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