Hyperplane separability and convexity of probabilistic point sets

We describe an $O(n^d)$ -time algorithm for computing the exact probability that two $d$-dimensional probabilistic point sets are linearly separable, for any fixed $d \geq 2$. A probabilistic point in $d$-space is a normal point, but with an associated probability of existence; the existence probabilities of all points are independent. We also show that the $d$-dimensional separability problem is equivalent to a $(d+1)$-dimensional convex hull membership problem, which asks for the probability that a query point lies inside the convex hull of $n$ probabilistic points. Using this reduction, we improve the current best bound for the convex hull membership problem by a factor of $n$. In addition, our algorithms can handle input degeneracies in which more than $k+1$ points may lie on a $k$-dimensional subspace, thus resolving an open problem in Agarwal et al 2013. Finally, we prove lower bounds for the separability problem via a reduction from the $k$-SUM problem, which show in particular that our $O(n^2)$ algorithms for $2$-dimensional separability and $3$-dimensional convex hull membership are nearly optimal.