On Pseudo-disk Hypergraphs

02/24/2018

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by   Boris Aronov, et al.

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Let F be a family of pseudo-disks in the plane, and P be a finite subset of F. Consider the hypergraph H(P,F) whose vertices are the pseudo-disks in P and the edges are all subsets of P of the form {D ∈ P | D ∩ S ≠∅}, where S is a pseudo-disk in F. We give an upper bound of O(nk^3) for the number of edges in H(P,F) of cardinality at most k. This generalizes a result of Buzaglo et al. (2013). As an application of our bound, we obtain an algorithm that computes a constant-factor approximation to the smallest _weighted_ dominating set in a collection of pseudo-disks in the plane, in expected polynomial time.