Cover-Decomposition and polychromatic numbers

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. © 2011 Springer-Verlag Berlin Heidelberg.