Advertisement

A theory of even functionals and their algorithmic applications

  • Jerzy W. Jaromczyk
  • Grzegorz Świcatek
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 700)

Abstract

We present a theory of even functionals of degree k. Even function als are homogeneous polynomials which are invariant with respect to permutations and reflections. These are evaluated on real symmetric matrices. Important examples of even functionais include functions for enumerating embeddings of graphs with k edges into a weighted graph with arbitrary (positive or negative) weights and computing kth moments (expected values of kth powers) of a binary form. This theory provides a uniform approach for evaluating even functionais and links their evaluation with expressions with matrices as operands. In particular, we show that any even functional of degree less than 7 can be computed in time O(nω), the time required to multiply two n × n matrices.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    A. V. Aho and J. E. Hopcroft and J. D. Ullman, The Design and Analysis of Computer Algorithms, Addison-Wesley, 1974.Google Scholar
  2. [2]
    D. Coppersmith and S. Winograd, Matrix multiplications via arithmetic progressions, in Proc. 19th Annual Acm Symp. on Theory of Computing, pp. 1–6, 1987.Google Scholar
  3. [3]
    P. Hansen, Methods of nonlinear 0–1 programming, Ann. Discrete Math., 5 (1979), pp. 53–70.Google Scholar
  4. [4]
    F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press (1973)Google Scholar
  5. [5]
    S. Kirkpatrick, C. D. Gelatt Jr., and M.P. Vecchi, Optimization by Simulated Annealing, Science, 220, 671–680, 1983.Google Scholar
  6. [6]
    J. van Leeuwen, Graph Algorithm, pp.526–631, in “Handbook of Theoretical Computer science”, Elsevier Science Publishers, 1990.Google Scholar
  7. [7]
    A. Schrijver, Theory of Linear and Integer Programming, John Wiley, New York, 1986.Google Scholar
  8. [8]
    J. Spencer Ten lectures on the probabilistic methods, SIAM, 1987.Google Scholar
  9. [9]
    B. L. van der Waerden, Modern algebra, Frederick Ungar Publishing Co., New York, 1950.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1993

Authors and Affiliations

  • Jerzy W. Jaromczyk
    • 1
  • Grzegorz Świcatek
    • 2
  1. 1.Department of Computer ScienceUniversity of KentuckyLexington
  2. 2.Department of MathematicsSUNY at Stony BrookStony Brook

Personalised recommendations