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 hyperedge-covers. In geo...

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Published inSIAM journal on discrete mathematics Vol. 27; no. 1; pp. 240 - 256
Main Authors Bollobás, Béla, Pritchard, David, Rothvoss, Thomas, Scott, Alex
Format Journal Article
LanguageEnglish
Published Philadelphia Society for Industrial and Applied Mathematics 01.01.2013
Subjects
Algorithms
Applied mathematics
Color
Coloring
Decomposition
Graphs
Linear programming
Mathematical analysis
Sensors
Trees
Upper bounds
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ISSN0895-4801
1095-7146
DOI10.1137/110856332

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Summary: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 hyperedge-covers. 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. [PUBLICATION ABSTRACT]
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ISSN:0895-4801
1095-7146
DOI:10.1137/110856332