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As Close as It Gets

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WALCOM: Algorithms and Computation (WALCOM 2016)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9627))

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Abstract

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.

M. Behrisch and G. Salzer— Supported by Austrian Science Fund (FWF) grant I836-N23.

M. Hermann— Supported by ANR-11-ISO2-003-01 Blanc International grant ALCOCLAN.

S. Mengel— Supported by QUALCOMM grant. Now at CRIL (UMR CNRS 8188), Lens, France.

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References

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Book  MATH  Google Scholar 

  3. Baker, K.A., Pixley, A.F.: Polynomial interpolation and the Chinese Remainder Theorem for algebraic systems. Mathematische Zeitschrift 143(2), 165–174 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  4. Behrisch, M., Hermann, M., Mengel, S., Salzer, G.: Minimal distance of propositional models (2015). abs/1502.06761

  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_56

    Chapter  Google Scholar 

  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)

    Article  Google Scholar 

  7. Böhler, E., Reith, S., Schnoor, H., Vollmer, H.: Bases for Boolean co-clones. Inf. Process. Lett. 96(2), 59–66 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Book  MATH  Google Scholar 

  9. Crescenzi, P., Rossi, G.: On the Hamming distance of constraint satisfaction problems. Theor. Comput. Sci. 288(1), 85–100 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  11. Jeavons, P., Cohen, D., Gyssens, M.: Closure properties of constraints. J. Assoc. Comput. Mach. 44(4), 527–548 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Chapter  Google Scholar 

  13. Lagerkvist, V.: Weak bases of Boolean co-clones. Inf. Process. Lett. 114(9), 462–468 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  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). http://dx.doi.org/10.1145/800133.804350

  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)

    Chapter  Google Scholar 

  16. Warshall, S.: A theorem on Boolean matrices. J. Assoc. Comput. Mach. 9(1), 11–12 (1962)

    Article  MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Technische Universität Wien, Vienna, Austria

    Mike Behrisch & Gernot Salzer

  2. LIX (UMR CNRS 7161), École Polytechnique, Palaiseau, France

    Miki Hermann & Stefan Mengel

Authors
  1. Mike Behrisch
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  2. Miki Hermann
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  3. Stefan Mengel
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  4. Gernot Salzer
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Corresponding author

Correspondence to Gernot Salzer .

Editor information

Editors and Affiliations

  1. Dept. of Computer Science and Engin., Bangladesh Univ. of Engin. & Technol., Dhaka, Bangladesh

    Mohammad Kaykobad

  2. Dept. of Computer Science, Sapienza University of Rome, Rome, Italy

    Rossella Petreschi

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Behrisch, M., Hermann, M., Mengel, S., Salzer, G. (2016). As Close as It Gets. In: Kaykobad, M., Petreschi, R. (eds) WALCOM: Algorithms and Computation. WALCOM 2016. Lecture Notes in Computer Science(), vol 9627. Springer, Cham. https://doi.org/10.1007/978-3-319-30139-6_18

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  • DOI: https://doi.org/10.1007/978-3-319-30139-6_18

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-30138-9

  • Online ISBN: 978-3-319-30139-6

  • eBook Packages: Computer ScienceComputer Science (R0)

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