Abstract
A colouring of a hypergraph’s vertices is polychromatic if every hyperedge contains at least one vertex of each colour; the polychromatic number is the maximum number of colours in such a colouring. Its dual, the cover-decomposition number, is the maximum number of disjoint hyperedge-covers. In geometric settings, there is extensive work on lower-bounding these numbers in terms of their trivial upper bounds (minimum hyperedge size & degree). Our goal is to get good lower bounds in natural hypergraph families not arising from geometry. We obtain algorithms yielding near-tight bounds for three hypergraph families: those with bounded hyperedge size, those representing paths in trees, and those with bounded VC-dimension. To do this, we link cover-decomposition to iterated relaxation of linear programs via discrepancy theory.
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Bollobás, B., Pritchard, D., Rothvoß, T., Scott, A. (2011). Cover-Decomposition and Polychromatic Numbers. In: Demetrescu, C., Halldórsson, M.M. (eds) Algorithms – ESA 2011. ESA 2011. Lecture Notes in Computer Science, vol 6942. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-23719-5_67
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DOI: https://doi.org/10.1007/978-3-642-23719-5_67
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