Hadwiger and Debrunner showed that for families of convex sets in
Rd with the property that among any p of them some q have a
common point, the whole family can be stabbed with pβq+1 points if pβ₯qβ₯d+1 and (dβ1)p<d(qβ1). 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((pβq+1)n4/3log8n(loglogn)1/3+np2) expected time. For polyhedra in R3, we
get an algorithm running in O((pβq+1)n5/2log10n(loglogn)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
RdΓZk or abstract convex geometries