Efficiently stabbing convex polygons and variants of the Hadwiger-Debrunner $(p, q)$-theorem

Hadwiger and Debrunner showed that for families of convex sets in $\mathbb{R}^d$ 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 $p \geq q \geq d+1$ 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 $O((p-q+1)n^{4/3}\log^{8} n(\log\log n)^{1/3} + np^2)$ expected time. For polyhedra in $\mathbb{R}^3$, we get an algorithm running in $O((p-q+1)n^{5/2}\log^{10} n(\log\log n)^{1/6} + np^3)$ 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 $\mathbb{R}^d\times\mathbb{Z}^k$ or abstract convex geometries.