Cover-decomposition and polychromatic numbers
A coloring of a hypergraph's vertices is polychromatic if every hyperedge contains at least one vertex of each color; the polychromatic number is the maximum number of colors in such a coloring. Its dual, the cover-decomposition number, is the maximum number of disjoint hyperedgecovers. In geometric hypergraphs, there is extensive work on lower-bounding these numbers in terms of their trivial upper bounds (minimum hyperedge size and degree); our goal here is to broaden the study beyond geometric settings. We obtain algorithms yielding near-tight bounds for three families of hypergraphs: bounded hyperedge size, paths in trees, and bounded Vapnik-Chervonenkis (VC)-dimension. This reveals that discrepancy theory and iterated linear program relaxation are useful for cover-decomposition. Finally, we discuss the generalization of cover-decomposition to sensor cover. © 2013 Society for Industrial and Applied Mathematics.
SIAM Journal on Discrete Mathematics
Bollobás, B., Pritchard, D., Rothvoss, T., & Scott, A. (2013). Cover-decomposition and polychromatic numbers. SIAM Journal on Discrete Mathematics, 27 (1), 240-256. https://doi.org/10.1137/110856332