Inapproximability Results for Equations over Finite Groups

  • Lars Engebretsen
  • Jonas Holmerin
  • Alexander Russell
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2380)

Abstract

An equation over a finite group G is an expression of form w1w2...wk = 1g, where each wi is a variable, an inverted variable, or a constant from G; such an equation is satisfiable if there is a setting of the variables to values in G so that the equality is realized. We study the problem of simultaneously satisfying a family of equations over a finite group G and show that it is NP-hard to approximate the number of simultaneously satisfiable equations to within |G| − ∈ for any ∈ > 0. This generalizes results of Håstad, who established similar bounds under the added condition that the group G is Abelian.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Sanjeev Arora, Carsten Lund, Rajeev Motwani, Madhu Sudan, and Márió Szegedy. Proof verification and the hardness of approximation problems. Journal of the ACM, 45(3):501–555, May 1998.Google Scholar
  2. 2.
    Sanjeev Arora and Shmuel Safra. Probabilistic checking of proofs: A new characterization of NP. Journal of the ACM, 45(1):70–122, January 1998.Google Scholar
  3. 3.
    David Mix Barrington, Pierre McKenzie, Cristopher Moore, Pascal Tesson, and Denis Thérien. Equation satisfiability and program satisfiability for finite monoids. In Proceedings of the 25th Annual Symposium on Mathematical Foundations of Computer Science, 2000.Google Scholar
  4. 4.
    David A. Mix Barrington, Howard Straubing, Denis Thérien. Non-uniform automata over groups. Information and Computation, 89(2): 109–132, 1990MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    François Bédard and Alain Goupil. The poset of conjugacy classes and decomposition of products in the symmetric group. Canad. Math. Bull., 35(2):152–160, 1992.MATHMathSciNetGoogle Scholar
  6. 6.
    Harry Dym and Henry P. McKean. Fourier Series and Integrals, volume 14 of Probability and Mathematical Statistics. Academic Press, 1972.Google Scholar
  7. 7.
    Uriel Feige. A threshold of ln n for approximating set cover. Journal of the ACM, 45(4):634–652, July 1998.Google Scholar
  8. 8.
    Uriel Feige, Shafi Goldwasser, László Lovász, Shmuel Safra, and Márió Szegedy. Interactive proofs and the hardness of approximating cliques. Journal of the ACM, 43(2):268–292, March 1996.Google Scholar
  9. 9.
    Harold Finkelstein and K. I. Mandelberg. On solutions of “equations in symmetric groups”. Jornal of Combinatorial Theory, Series A, 25(2):142–152, 1978.MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Mikael Goldmann and Alexander Russell. The computational complexity of solving systems of equations over finite groups. In Proceedings of the Fourteenth Annual IEEE Conference on Computational Complexity, Atlanta, GA, May 1999.Google Scholar
  11. 11.
    Johan Håstad. Some optimal inapproximability results. Journal of the ACM, 48(4):798–859, July 2001.Google Scholar
  12. 12.
    Christos H. Papadimitriou and Mihalis Yannakakis. Optimization, approximation, and complexity classes. Journal of Computer and System Sciences, 43(3):425–440, December 1991.Google Scholar
  13. 13.
    Ran Raz. A parallel repetition theorem. SIAM Journal on Computing, 27(3):763–803, June 1998.Google Scholar
  14. 14.
    Jean-Pierre Serre. Linear Representations of Finite Groups, volume 42 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1977.Google Scholar
  15. 15.
    Arnold Schönhage and Volker Strassen. Schnelle Multiplikation großer Zahlen. Computing, 7:281–292, 1971.MATHCrossRefGoogle Scholar
  16. 16.
    Richard P. Stanley. Enumerative Combinatorics, Volume 2, volume 62 of Cambridge Studies in Advances Mathematics. Cambridge University Press, 1999.Google Scholar
  17. 17.
    Sergej P. Strunkov. On the theory of equations on finite groups. Izvestija Rossijskoj Akademii Nauk, Serija Matematičeskaja, 59(6):171–180, 1995.MathSciNetGoogle Scholar
  18. 18.
    Audrey Terras. Fourier Analysis on Finite Groups and Applications, volume 43 of London Mathematical Society student texts. Cambridge University Press, Cambridge, 1999.Google Scholar
  19. 19.
    Uri Zwick. Approximation algorithms for constraint satisfaction problems involving at most three variables per constraint. In Proceedings of the Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 201–210, San Francisco, California, 25–27 January 1998.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Lars Engebretsen
    • 1
  • Jonas Holmerin
    • 1
  • Alexander Russell
    • 2
  1. 1.Department of Numerical Analysis and Computer ScienceRoyal Institute of TechnologyStockholmSweden
  2. 2.Department of Computer Science and EngineeringUniversity of ConnecticutStorrs

Personalised recommendations