Efficiently stabbing convex polygons and variants of the Hadwiger-Debrunner (p, q)-theorem
share
Hadwiger and Debrunner showed that for families of convex sets in ℝ^d with 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 n convex polygons of constant size in the plane in O((p-q+1)n^4/3log^2+ϵ(n) + p^2log(p)) expected time. For convex polyhedra in ℝ^3, the method yields an algorithm running in O((p-q+1)n^13/5+ϵ + p^4) expected time. We also show that analogous results of the Hadwiger and Debrunner (p,q)-theorem hold in other settings, such as convex sets in ℝ^d×ℤ^k or abstract convex geometries.
READ FULL TEXT