Advertisement

An upper bound for the maximum cut mean value

  • Alberto Bertoni
  • Paola Campadelli
  • Roberto Posenato
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1335)

Abstract

Let MaxCut(G) be the value of the maximum cut of a graph G. Let f(x, n) be the expectation of MaxCut(G)/xn for random graphs with n vertices and xn edges and let r(x,n) be the expectation of MaxCut(G)/xn for random 2x-regular graphs with n vertices. We prove, for sufficiently large x:
$$\lim _{n \to \infty } f(x,n) \leqslant \frac{1}{2} + \sqrt {\frac{{\ln 2}}{{2x}}} ,$$
(1)
$$\lim _{n \to \infty } r(x,n) \leqslant \frac{1}{2} + \frac{1}{{\sqrt x }} + \frac{1}{2}\frac{{\ln x}}{x}.$$
(1)

Keywords

Random Graph Maximum Clique Maximum Clique Problem Hopfield Network Small Size Instance 
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.
    Francisco Barahona, M. Jünger, M. Grötschel, and G. Reinelt. An application of combinatorial optimization to statistical physics and circuit layout design. Operations Research, (36):493–513, 1988.Google Scholar
  2. 2.
    Joel Friedman. On the second eigenvalue and random walks in random d-regular graphs. Combinatorica, 11(4):331–365, 1991.Google Scholar
  3. 3.
    Michel X. Goemans and David P. Williamson..878-approximation algorithms for MAX CUT and MAX 2SAT. In Proceedings of the 26th ACM Symposium on the Theory of Computation, 1994.Google Scholar
  4. 4.
    F. Hadlock. Finding a maximum cut of a planar graph in polynomial time. SIAM Journal on Computing, 4(3):221–225, September 1975.Google Scholar
  5. 5.
    Arun Jagota. Approximating maximum clique with a Hopfield network. IEEE Transactions on Neural Networks, 6(3):724–735, May 1995.Google Scholar
  6. 6.
    Richard M. Karp. Reducibility among combinatorial problems. In R. E. Miller and J. W. Thatcher, editors, Complexity of Computer Computations, pages 85–103, New York, 1972. Plenum Press.Google Scholar
  7. 7.
    J. Komlos and R. Paturi. Effect of conectivity in associative memory models. Technical Report CS88-131, University of California. San Diego, August 1988.Google Scholar
  8. 8.
    Sartaj Sahni and Teofilo Gonzalez. P-Complete Approximation Problems. Journal of the ACM, 23(3):555–565, July 1976.Google Scholar
  9. 9.
    Johan Håstad. Some optimal in-approximability results. November 1996.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1997

Authors and Affiliations

  • Alberto Bertoni
    • 1
  • Paola Campadelli
    • 1
  • Roberto Posenato
    • 1
  1. 1.Dipartimento di Scienze dell'InformazioneUniversità degli Studi di MilanoMilanoItaly

Personalised recommendations