Efficiently Stabbing Convex Polygons and Variants of the Hadwiger-Debrunner (p, q)-Theorem
Hadwiger and Debrunner showed that for families of convex sets in Rdwith the property that among any p of them some q have a common point, the whole family can be stabbed with p − q + 1 points if p ≥ q ≥ d + 1 and (d − 1)p < d(q − 1). This generalizes a classical result by Helly. We show how such a stabbing set can be computed for a family of convex polygons in the plane with a total of n vertices in O((p − q + 1)n4/3log8n( log log n)1/3+ np2) expected time. For polyhedra in R3, we get an algorithm running in O((p − q + 1)n5/2log10n(log log n)1/6+ np3) expected time. We also investigate other conditions on convex polygons for which our algorithm can find a fixed number of points stabbing them. Finally, we show that analogous results of the Hadwiger and Debrunner (p, q)-theorem hold in other settings, such as convex sets in Rd× Zkor abstract convex geometries.
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Copyright (c) 2022 Justin Dallant, Patrick Schnider
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