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CoverDecomposition and Polychromatic Numbers
"... A colouring of a hypergraph’s vertices is polychromatic if every hyperedge contains at leastone vertex ofeach colour; the polychromatic number is the maximum number of colours in such a colouring. Its dual, the coverdecomposition number, is the maximum number of disjoint hyperedge covers. In geomet ..."
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A colouring of a hypergraph’s vertices is polychromatic if every hyperedge contains at leastone vertex ofeach colour; the polychromatic number is the maximum number of colours in such a colouring. Its dual, the coverdecomposition number, is the maximum number of disjoint hyperedge covers. In geometric hypergraphs, there is extensive work on lowerbounding 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 neartight bounds for three families of hypergraphs: bounded hyperedge size, paths in trees, and bounded VCdimension. This reveals that discrepancy theory and iterated linear program relaxation are useful for coverdecomposition. Finally, we discuss the generalization of coverdecomposition to sensor cover.
εNets for Halfspaces Revisited∗
, 2014
"... “It is a damn poor mind indeed which can’t think of at least two ways to spell any word.” – Andrew Jackson Given a set P of n points in R3, we show that, for any ε> 0, there exists an εnet of P for halfspace ranges, of size O(1/ε). We give five proofs of this result, which are arguably simpler t ..."
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“It is a damn poor mind indeed which can’t think of at least two ways to spell any word.” – Andrew Jackson Given a set P of n points in R3, we show that, for any ε> 0, there exists an εnet of P for halfspace ranges, of size O(1/ε). We give five proofs of this result, which are arguably simpler than previous proofs [?,?,?]. We also consider several related variants of this result, including the case of points and pseudodisks in the plane. 1