On the Shortest Separating Cycle
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According to a result of Arkin (2016), given n point pairs in the plane, there exists a simple polygonal cycle that separates the two points in each pair to different sides; moreover, a O(√(n))-factor approximation with respect to the minimum length can be computed in polynomial time. Here the following results are obtained: (I) We extend the problem to geometric hypergraphs and obtain the following characterization of feasibility. Given a geometric hypergraph on points in the plane with hyperedges of size at least 2, there exists a simple polygonal cycle that separates each hyperedge if and only if the hypergraph is 2-colorable. (II) We extend the O(√(n))-factor approximation in the length measure as follows: Given a geometric graph G=(V,E), a separating cycle (if it exists) can be computed in O(m+ nlogn) time, where |V|=n, |E|=m. Moreover, a O(√(n))-approximation of the shortest separating cycle can be found in polynomial time. Given a geometric graph G=(V,E) in R^3, a separating polyhedron (if it exists) can be found in O(m+ nlogn) time, where |V|=n, |E|=m. Moreover, a O(n^2/3)-approximation of a separating polyhedron of minimum perimeter can be found in polynomial time. (III) Given a set of n point pairs in convex position in the plane, we show that a (1+ε)-approximation of a shortest separating cycle can be computed in time n^O(ε^-1/2). In this regard, we prove a lemma on convex polygon approximation that is of independent interest.
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