Typical rounding problems

Abstract

The linear discrepancy problem is to round a given [0, 1]–vector x to a binary vector y such that the rounding error with respect to a linear form is small, i.e., such that ‖A(x−y)‖∞is small for some given matrix A. The discrepancy problem is the special case of x = (½,…, ½). A famous result of Beck and Spencer (1984) as well as Lovász, Spencer and Vesztergombi (1986) shows that the linear discrepancy problem is not much harder than this special case: Any linear discrepancy problem can be solved with at most twice the maximum rounding error among the discrepancy problems of the submatrices of A. In this paper we strengthen this result for the common situation that the discrepancy of submatrices having n0 columns is bounded by Cnα0 for some C > 0, α ∈(0, 1]. In this case, we improve the constant by which the general problem is harder than the discrepancy one, down to 2(⅔)α. We also find that a random vector x has expected linear discrepancy 2(½)αCnαonly. Hence in the typical situation that the discrepancy is decreasing for smaller matrices, the linear discrepancy problem is even less difficult compared to the discrepancy one than assured by the results of Beck and Spencer and Lovász, Spencer and Vesztergombi.