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
The discrepancy method: randomness and complexity
The discrepancy method: randomness and complexity
Guessing secrets with inner product questions
SODA '02 Proceedings of the thirteenth annual ACM-SIAM symposium on Discrete algorithms
Guessing secrets efficiently via list decoding
SODA '02 Proceedings of the thirteenth annual ACM-SIAM symposium on Discrete algorithms
Searching games with errors---fifty years of coping with liars
Theoretical Computer Science
Combinatorics, Probability and Computing
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 ||A(x - y)||∞ is small for some given matrix A. The combinatorial discrepancy problem is the special case of x=(1/2,...,1/2)t. A famous result of Beck and Spencer [Math. Programming 30 (1984) 88] as well as Lovász et al. [European J. Combin. 7 (1986) 151] 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 Cn0α 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 from 2 down to 2(2/3)α. We also find that a random vector has expected linear discrepancy 2(1/2)αCnα 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 previous results. We also obtain the bound lindisc(A,x) ≤ 2(2α/(21-α- 1))C||x||1α. Our proofs use a reduction to Pusher-Chooser games.