Discrepancy of set-systems and matrices
European Journal of Combinatorics
Algorithmic Chernoff-Hoeffding inequalities in integer programming
Random Structures & Algorithms
Handbook of combinatorics (vol. 2)
The cost of the missing bit: communication complexity with help
STOC '98 Proceedings of the thirtieth annual ACM symposium on Theory of computing
The discrepancy method: randomness and complexity
The discrepancy method: randomness and complexity
Approximation of Multi-color Discrepancy
RANDOM-APPROX '99 Proceedings of the Third International Workshop on Approximation Algorithms for Combinatorial Optimization Problems: Randomization, Approximation, and Combinatorial Algorithms and Techniques
Balanced Coloring: Equally Easy for All Numbers of Colors?
STACS '02 Proceedings of the 19th Annual Symposium on Theoretical Aspects of Computer Science
APPROX '02 Proceedings of the 5th International Workshop on Approximation Algorithms for Combinatorial Optimization
Structured Randomized Rounding and Coloring
FCT '01 Proceedings of the 13th International Symposium on Fundamentals of Computation Theory
Discrepancy of (centered) arithmetic progressions in Zp
European Journal of Combinatorics
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We investigate a refined recursive coloring approach to construct balanced colorings for hypergraphs. A coloring is called balanced if each hyperedge has (roughly) the same number of vertices in each color. We provide a recursive randomized algorithm that colors an arbitrary hypergraph (n vertices, m edges) with c colors with discrepancy at most O(√n/c logm). The algorithm has expected running time O(nm log c). This result improves the bound of O(√n log(cm)) achieved with probability at least 1/2 by a random coloring that independently chooses a random color for each vertex (fair dice coloring). Our approach also lowers the current best upper bound for the c-color discrepancy in the case n = m to O(√n/c log c) and extends the algorithm of Matoušek, Welzl and Wernisch for hypergraphs having bounded dual shatter function to arbitrary numbers of colors.