Israel Journal of Mathematics

, Volume 158, Issue 1, pp 159–191 | Cite as

Random weighting, asymptotic counting, and inverse isoperimetry

  • Alexander Barvinok
  • Alex Samorodnitsky
Article
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Abstract

For a family X of k-subsets of the set {1, …, n}, let |X| be the cardinality of X and let Γ(X, μ) be the expected maximum weight of a subset from X when the weights of 1, …, n are chosen independently at random from a symmetric probability distribution μ on ℝ. We consider the inverse isoperimetric problem of finding μ for which Γ(X, μ) gives the best estimate of ln |X|. We prove that the optimal choice of μ is the logistic distribution, in which case Γ(X, μ) provides an asymptotically tight estimate of ln |X| as k −1 ln |X} grows. Since in many important cases Γ(X, μ) can be easily computed, we obtain computationally efficient approximation algorithms for a variety of counting problems. Given μ, we describe families X of a given cardinality with the minimum value of Γ(X, μ), thus extending and sharpening various isoperimetric inequalities in the Boolean cube.

Keywords

Cumulative Distribution Function Independent Random Variable Isoperimetric Inequality Moment Generate Function Logistic Distribution 
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.
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Copyright information

© The Hebrew University of Jerusalem 2007

Authors and Affiliations

  • Alexander Barvinok
    • 1
  • Alex Samorodnitsky
    • 2
  1. 1.Department of MathematicsUniversity of MichiganAnn ArborUSA
  2. 2.Department of Computer ScienceThe Hebrew University of JerusalemGivat Ram, JerusalemIsrael

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