Stabbing Convex Polygons with a Segment or a Polygon

Let $\mathcal{O} = \{O_1, \ldots, O_m\}$ be a set of mconvex polygons in 茂戮驴2with a total of nvertices, and let Bbe another convex k-gon. A placementof B, any congruent copy of B(without reflection), is called freeif Bdoes not intersect the interior of any polygon in $\mathcal{O}$ at this placement. A placement zof Bis called criticalif Bforms three "distinct" contacts with $\mathcal{O}$ at z. Let $\varphi(B, \mathcal{O})$ be the number of free critical placements. A set of placements of Bis called a stabbing setof $\mathcal{O}$ if each polygon in $\mathcal{O}$ intersects at least one placement of Bin this set.We develop efficient Monte Carlo algorithms that compute a stabbing set of size h= O(h*logm), with high probability, where h*is the size of the optimal stabbing set of $\mathcal{O}$. We also improve bounds on $\varphi(B, \mathcal{O})$ for the following three cases, namely, (i) Bis a line segment and the obstacles in $\mathcal{O}$ are pairwise-disjoint, (ii) Bis a line segment and the obstacles in $\mathcal{O}$ may intersect (iii) Bis a convex k-gon and the obstacles in $\mathcal{O}$ are disjoint, and use these improved bounds to analyze the running time of our stabbing-set algorithm.