Israel Journal of Mathematics

, Volume 158, Issue 1, pp 159–191

Random weighting, asymptotic counting, and inverse isoperimetry

  • Alexander Barvinok
  • Alex Samorodnitsky
Article

DOI: 10.1007/s11856-007-0008-8

Cite this article as:
Barvinok, A. & Samorodnitsky, A. Isr. J. Math. (2007) 158: 159. doi:10.1007/s11856-007-0008-8

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.

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