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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.
Making Octants Colorful and Related Covering Decomposition Problems
 SIAM J. Discrete Math
, 1948
"... Abstract. We give new positive results on the longstanding open problem of geometric covering decomposition for homothetic polygons. In particular, we prove that for any positive integer k, every finite set of points in R3 can be colored with k colors so that every translate of the negative octant ..."
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Abstract. We give new positive results on the longstanding open problem of geometric covering decomposition for homothetic polygons. In particular, we prove that for any positive integer k, every finite set of points in R3 can be colored with k colors so that every translate of the negative octant containing at least k6 points contains at least one of each color. The best previously known bound was doubly exponential in k. This yields, among other corollaries, the first polynomial bound for the decomposability of multiple coverings by homothetic triangles. We also investigate related decomposition problems involving intervals appearing on a line. We prove that no algorithm can dynamically maintain a decomposition of a multiple covering by intervals under insertion of new intervals, even in a semionline model, in which some coloring decisions can be delayed. This implies that a wide range of sweeping plane algorithms cannot guarantee any bound even for special cases of the octant problem.
Journal of Computational Geometry jocg.org MAKING TRIANGLES COLORFUL∗
"... Abstract. We prove that for any finite point set P in the plane, a triangle T, and a positive integer k, there exists a coloring of P with k colors such that any homothetic copy of T containing at least 144k8 points of P contains at least one of each color. This is the first polynomial bound for ran ..."
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Abstract. We prove that for any finite point set P in the plane, a triangle T, and a positive integer k, there exists a coloring of P with k colors such that any homothetic copy of T containing at least 144k8 points of P contains at least one of each color. This is the first polynomial bound for range spaces induced by homothetic polygons. The only previously known bound for this problem applies to the more general case of octants in R3, but is doubly exponential. 1