On the Expected Diameter, Width, and Complexity of a Stochastic Convex-Hull

We investigate several computational problems related to stochastic convex hull (SCH). Given a stochastic dataset consisting of n points in R each of which has an existence probability, a SCH refers to the convex hull of a realization of the dataset, i.e., a random sample including each point with its existence probability. We are interested in the computation of some expected statistics of a SCH, including diameter, width, and combinatorial complexity. For diameter, we establish the first deterministic 1.633-approximation algorithm with a time complexity polynomial in both n and d. For width, two approximation algorithms are provided: a deterministic O(1)-approximation running in O(n logn) time, and a fully polynomial-time randomized approximation scheme (FPRAS). For combinatorial complexity, we propose an exact O(n)-time algorithm. Our solutions exploit many geometric insights in Euclidean space, some of which might be of independent interest.