# Algorithmic Chernoff-Hoeffding inequalities in integer programming

## Abstract

- 1.
Are there more examples of NP-hard combinatorial optimization problems for which derandomization yields constant factor approximations in polynomial-time ?

- 2.
The pessimistic estimator technique shows an

*O*(*mn*)-time implementation of the conditional probability method on the RAM model of computation in case of*m*large deviation events associated to*m unweighted*sums of*n*indepependent Bernoulli trials. Is there a fast algorithm also in case of*rational*weighted sums of Bernoulli trials ?

This paper is a contribution to both problems. On the one hand we will present the first implementation of the conditional probability method on the RAM model of computation for *rational* weighted sums of Bernoulli trials, which needs *O*(*mn*^{2} log *mn/ε*)-time, where *ε* is the success probability of the given *m* events. Hence the present gap between the running times of the unweighted and the weighted conditional probability method is a *O*(*n* log *mn*/*ε*)-factor.

On the other hand we will show, applying such derandomization procedures, which we call the algorithmic Chernoff-Hoeffding inequalities, the first constant factor approximation algorithms for maximizing integer programs with rational constraint matrix and integer domains, provided that the entries of the constraint vector grow logarithmically in the number of constraints.

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