Discrepancy of set-systems and matrices
European Journal of Combinatorics
Randomization, derandomization and antirandomization: three games
Theoretical Computer Science
Handbook of combinatorics (vol. 2)
Lattice approximation and linear discrepency of totally unimodular matrices
SODA '01 Proceedings of the twelfth annual ACM-SIAM symposium on Discrete algorithms
Recursive Randomized Coloring Beats Fair Dice Random Colorings
STACS '01 Proceedings of the 18th Annual Symposium on Theoretical Aspects of Computer Science
Linear and Hereditary Discrepancy
Combinatorics, Probability and Computing
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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 驴A(x-y)驴驴 is small for some given matrix A. The discrepancy problem is the special case of x = (1/2, . . . , 1/2). A famous result of Beck and Spencer (1984) as well as Lov谩sz, 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 Cn0驴for some C 0, 驴 驴 (0, 1). In this case, we improve the constant by which the general problem is harder than the discrepancy one, down to 2(2/3)驴. We also find that a random vector x has expected linear discrepancy 2(1/2)驴Cn驴 only. 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 Lov谩sz, Spencer and Vesztergombi.