Advertisement

Faster Algebraic Algorithms for Path and Packing Problems

  • Ioannis Koutis
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5125)

Abstract

We study the problem of deciding whether an n-variate polynomial, presented as an arithmetic circuit G, contains a degree k square-free term with an odd coefficient. We show that if G can be evaluated over the integers modulo 2k + 1 in time t and space s, the problem can be decided with constant probability in O((kn + t)2 k ) time and O(kn + s) space. Based on this, we present new and faster algorithms for two well studied problems: (i) an O *(2 mk ) algorithm for the m-set k-packing problem and (ii) an O *(23k/2) algorithm for the simple k-path problem, or an O *(2 k ) algorithm if the graph has an induced k-subgraph with an odd number of Hamiltonian paths. Our algorithms use poly(n) random bits, comparing to the 2O(k) random bits required in prior algorithms, while having similar low space requirements.

Keywords

Packing Problem Hamiltonian Path Deterministic Algorithm Simple Path Constant Probability 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Alon, N., Yuster, R., Zwick, U.: Color coding. Journal of the ACM 42(4), 844–856 (1995)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Chen, J., Lu, S., Sze, S.-H., Zhang, F.: Improved algorithms for path, matching, and packing problems. In: SODA 2007: Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms, Philadelphia, PA, USA. Society for Industrial and Applied Mathematics, pp. 298–307 (2007)Google Scholar
  3. 3.
    Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer, Heidelberg (1999)CrossRefMATHGoogle Scholar
  4. 4.
    Fellows, M.R., Knauer, C., Nishimura, N., Ragde, P., Rosamond, F.A., Stege, U., Thilikos, D.M., Whitesides, S.: Faster fixed-parameter tractable algorithms for matching and packing problems. In: Albers, S., Radzik, T. (eds.) ESA 2004. LNCS, vol. 3221, pp. 311–322. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  5. 5.
    Jia, W., Zhang, C., Chen, J.: An efficient parameterized algorithm for m-set packing. J. Algorithms 50(1), 106–117 (2004)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Kneis, J., Mölle, D., Richter, S., Rossmanith, P.: Divide-and-color. In: WG: Graph-Theoretic Concepts in Computer Science, 32nd International Workshop, pp. 58–67 (2006)Google Scholar
  7. 7.
    Koutis, I.: A faster parameterized algorithm for set packing. Information Processing Letters 94(1), 4–7 (2005)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Liu, Y., Lu, S., Chen, J., Sze, S.-H.: Greedy localization and color-coding: Improved matching and packing algorithms. In: Bodlaender, H.L., Langston, M.A. (eds.) IWPEC 2006. LNCS, vol. 4169, pp. 84–95. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  9. 9.
    Papadimitriou, C.H., Yannakakis, M.: On limited nondeterminism and the complexity of the V-C dimension. J. Comput. Syst. Sci. 53(2), 161–170 (1996)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Terras, A.: Fourier Analysis on Finite Groups and Applications. Cambridge University, Cambridge (1999)CrossRefMATHGoogle Scholar
  11. 11.
    Valiant, L.G.: Why is boolean complexity difficult? In: Boolean Function Complexity. Lond. Math. Soc. Lecure Note Ser, vol. 169Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Ioannis Koutis
    • 1
  1. 1.Computer Science DepartmentCarnegie Mellon UniversityPittsburgh

Personalised recommendations