Recent progress on combinatorics and algorithms for low discrepancy roundings

Given a [0,1]-valued n-dimensional vector a = (a ₁, a ₂, . . . , a _n )∈ [0,1] ^V indexed by a set V = {v ₁, v ₂, . . . , v _n }, we consider the problem of approximating a by a binary (i.e., {0,1}-valued) vector α = (α ₁, α ₂, . . . , α _n ) {0,1} ^V under the discrepancy measure with respect to a hypergraph H = (V, F). We are interested in the properties of low-discrepancy roundings. Especially, we survey recent works on the combinatorial properties of a global rounding; that is, rounding whose discrepancy is less than 1.