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Algorithmic Chernoff-Hoeffding inequalities in integer programming
Random Structures & Algorithms
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STOC '98 Proceedings of the thirtieth annual ACM symposium on Theory of computing
Balancing vectors and Gaussian measures of n-dimensional convex bodies
Random Structures & Algorithms
SODA '97 Proceedings of the eighth annual ACM-SIAM symposium on Discrete algorithms
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On the linear and hereditary discrepancies
European Journal of Combinatorics
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
Dispencery in different numbers of colors
Discrete Mathematics
Balanced Coloring: Equally Easy for All Numbers of Colors?
STACS '02 Proceedings of the 19th Annual Symposium on Theoretical Aspects of Computer Science
Linear and Hereditary Discrepancy
Combinatorics, Probability and Computing
Theoretical Computer Science
Non-independent randomized rounding and coloring
Discrete Applied Mathematics - Special issue: Efficient algorithms
Improved bounds and schemes for the declustering problem
Theoretical Computer Science
Latin squares and low discrepancy allocation of two-dimensional data
European Journal of Combinatorics
Non-independent randomized rounding and coloring
Discrete Applied Mathematics - Special issue: Efficient algorithms
Hereditary Discrepancies in Different Numbers of Colors II
SIAM Journal on Discrete Mathematics
The entropy rounding method in approximation algorithms
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
Rounding of sequences and matrices, with applications
WAOA'05 Proceedings of the Third international conference on Approximation and Online Algorithms
Roundings respecting hard constraints
STACS'05 Proceedings of the 22nd annual conference on Theoretical Aspects of Computer Science
Discrepancy of (centered) arithmetic progressions in Zp
European Journal of Combinatorics
Random partitions and edge-disjoint Hamiltonian cycles
Journal of Combinatorial Theory Series B
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In this article we introduce combinatorial multicolour discrepancies and generalize several classical results from $2$-colour discrepancy theory to $c$ colours ($c \geq 2$). We give a recursive method that constructs $c$-colourings from approximations of $2$-colour discrepancies. This method works for a large class of theorems, such as the ‘six standard deviations’ theorem of Spencer (1985), the Beck–Fiala (1981) theorem, the results of Matoušek, Wernisch and Welzl (1994) and Matoušek (1995) for bounded VC-dimension, and Matoušek and Spencer's (1996) upper bound for the arithmetic progressions. In particular, the $c$-colour discrepancy of an arbitrary hypergraph ($n$ vertices, $m$ hyperedges) is \[ \OO\Bigl(\sqrt{\tfrac n c\,\log m}\Bigr). \] If $m = \OO(n)$, then this bound improves to \[ \OO\Bigl(\sqrt{\tfrac n c\,\log c}\Bigr). \]On the other hand there are examples showing that discrepancy in $c$ colours can not be bounded in terms of two-colour discrepancies in general, even if $c$ is a power of 2. For the linear discrepancy version of the Beck–Fiala theorem, the recursive approach also fails.Here we extend the method of floating colours via tensor products of matrices to multicolourings, and prove multicolour versions of the Beck–Fiala theorem and the Bárány–Grinberg theorem. Using properties of the tensor product we derive a lower bound for the $c$-colour discrepancy of general hypergraphs. For the hypergraph of arithmetic progressions in $\{1, \ldots, n\}$ this yields a lower bound of $\frac{1}{25 \sqrt c} \sqrt[4]{n}$ for the discrepancy in $c$ colours. The recursive method shows an upper bound of $\OO(c^{-0.16} \sqrt[4]{n})$