Algorithmic Chernoo-hoeeding Inequalities in Integer Programming Zz

Proofs of classical Chernoo-Hoeeding bounds have been used to obtain polynomial time implementations of Spencer's derandomization method of conditional probabilities on usual nite machine models: given m events whose complements are large deviations corresponding to weighted sums of n mutually independent Bernoulli trials, Raghavan's lattice approximation algorithm constructs for 0 ? 1 weights and integer deviation terms in O(mn)-time a point for which all events hold. For rational weighted sums of Bernoulli trials the lattice approximation algorithm or Spencer's hyperbolic cosine algorithm are deterministic procedures, but a polynomial-time implementation was not known. We resolve this problem with an O(mn 2 log mn)-time algorithm, whenever the probability that all events hold is at least > 0. Since such algorithms simulate the proof of the underlying large deviation inequality in a constructive way, we call it the algorithmic version of the inequality. Applications to general packing integer programs and resource constrained scheduling result in tight and polynomial-time approximations algorithms.