Discrepancy of set-systems and matrices
European Journal of Combinatorics
Chernoff-Hoeffding Bounds for Applications with Limited Independence
SIAM Journal on Discrete Mathematics
The cost of the missing bit: communication complexity with help
STOC '98 Proceedings of the thirtieth annual ACM symposium on Theory of computing
An extension of the Lovász local lemma, and its applications to integer programming
Proceedings of the seventh annual ACM-SIAM symposium on Discrete algorithms
The discrepancy method: randomness and complexity
The discrepancy method: randomness and complexity
Recursive Randomized Coloring Beats Fair Dice Random Colorings
STACS '01 Proceedings of the 18th Annual Symposium on Theoretical Aspects of Computer Science
Non-independent Randomized Rounding and an Application to Digital Halftoning
ESA '02 Proceedings of the 10th Annual European Symposium on Algorithms
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In this paper we propose an advanced randomized coloring algorithm for the problem of balanced colorings of hypergraphs (discrepancy problem). It allows to use structural information about the hypergraph in the design of the random experiment. This yields colorings having smaller discrepancy than those independently coloring the vertices. We also obtain more information about the coloring, or, conversely, we may enforce the random coloring to have special properties. Due to the dependencies, these random colorings need fewer random bits to be constructed, and computing their discrepancy can be done faster. We apply our method to hypergraphs of d-dimensional boxes. Among others, we observe a factor 2d/2 decrease in discrepancy and a reduction of the number of random bits needed by a factor of 2d. Since the discrepancy problem is a particular rounding problem, our approach is a randomized rounding strategy for the corresponding ILP-relaxation that beats the usual randomized rounding.