On the minimum edge size for 2-colorability and realizability of hypergraphs by axis-parallel rectangles


Given a hypergraph H = (V, E) what is the minimum integer λ(H) such that the sub-hypergraph with edges of size at least λ(H) is 2-colorable? We consider the computational problem of finding the smallest such integer for a given hypergraph, and show that it is NP-hard to approximate it to within log m where m = |E|. For most geometric hypergraphs, i.e., those defined on a set of n points by intersecting it with some shapes, it is well known that there is a coloring with 2 colors 'red' and 'blue', such that any hyperedge containing c log n points, for some constant c, is bi-chromatic, i.e., contains points of both colors. We observe that indeed, for several such hypergraph families, this is the best possible - i.e., there are some n points where there will always be a hyperedge with Ω(log n) points that is mono-chromatic. These results follow from results on the indecomposability of coverings. We also show that a frequently used hypergraph, used in the literature on indecomposable coverings cannot be realized by axis-parallel rectangles in the plane. This problem was mentioned in a paper of Pach et al. on indecomposable coverings.

Publication Title

CCCG 2017 - 29th Canadian Conference on Computational Geometry, Proceedings

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