Randomly rounding rationals with cardinality constraints and derandomizations

Abstract

We show how to generate randomized roundings of rational vectors that satisfy hard cardinality constraints and allow large deviations bounds. This improves and extends earlier results by Srinivasan (FOCS 2001), Gandhi et al. (FOCS 2002) and the author (STACS 2006). Roughly speaking, we show that also for rounding arbitrary rational vectors randomly or deterministically, it suffices to understand the problem for {0, 1/2} vectors (which typically is much easier). So far, this was only known for vectors with entries in 2 -ℓℤ, ℓ ∈ ℕ. To prove the general case, we exhibit a number of results of independent interest, in particular, a quite useful lemma on negatively correlated random variables, an extension of de Werra's (RAIRO 1971) coloring result for unimodular hypergraphs and a sufficient condition for a unimodular hypergraph to have a perfectly balanced non-trivial partial coloring. We also show a new solution for the general derandomization problem for rational matrices. © Springer-Verlag Berlin Heidelberg 2007.