A Gambling Game Arising in the Analysis of Adaptive Randomized Rounding

  • Richard M. Karp
  • Claire Kenyon
Conference paper

DOI: 10.1007/978-3-540-45198-3_28

Part of the Lecture Notes in Computer Science book series (LNCS, volume 2764)
Cite this paper as:
Karp R.M., Kenyon C. (2003) A Gambling Game Arising in the Analysis of Adaptive Randomized Rounding. In: Arora S., Jansen K., Rolim J.D.P., Sahai A. (eds) Approximation, Randomization, and Combinatorial Optimization.. Algorithms and Techniques. Lecture Notes in Computer Science, vol 2764. Springer, Berlin, Heidelberg

Abstract

Let y be a positive real number and let {Xi} be an infinite sequence of Bernoulli random variables with the following property: in every realization of the random variables, \(\sum_{i=1}^{\infty} E[X_i|X_1,X_2,\cdots, X_{i-1}] \leq y\). We specify a function F(x,y) such that, for every positive integer x and every positive real y, \(P(\sum_{i=1}^{\infty} X_i \geq x) \leq F(x,y)\); moreover, for every x and y, F(x,y) is the best possible upper bound. We give an interpretation of this stochastic process as a gambling game, characterize optimal play in this game, and explain how our results can be applied to the analysis of multi-stage randomized rounding algorithms, giving stronger results than can be obtained using the traditional Hoeffding bounds and martingale tail inequalities.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Richard M. Karp
    • 1
  • Claire Kenyon
    • 2
  1. 1.UC Berkeley and ICSI 
  2. 2.Ecole Polytechnique and IUF 

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