We investigate several computational problems related to  stochastic convex hull (SCH). Given a stochastic dataset consisting of n points in R^d 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^{d+1} \log n) time, and a fully polynomial-time randomized approximation scheme (FPRAS). For combinatorial complexity, we propose an exact O(n^d)-time algorithm. Our solutions exploit many geometric insights in Euclidean space, some of which might be of independent interest.
We investigate several computational problems related to the stochastic convex hull (SCH). Given a stochastic dataset consisting of n points in R^d 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 computing certain 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^{d+1} \log n) time, and a fully polynomial-time randomized approximation scheme (FPRAS). For combinatorial complexity, we propose an exact O(n^d)-time algorithm. Our solutions exploit many geometric insights in Euclidean space, some of which might be of independent interest.
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