Approximation Algorithms for Radius-Based, Two-Stage Stochastic Clustering Problems with Budget Constraints

The main focus of this paper is radius-based clustering problems in the two-stage stochastic setting with recourse, where the inherent stochasticity of the model comes in the form of a budget constraint. We also explore a number of variants where additional constraints are imposed on the first-stage decisions, specifically matroid and multi-knapsack constraints. Further, we show that our problems have natural applications to allocating healthcare testing centers. The eventual goal is to provide results for supplier-like problems in the most general distributional setting, where there is only black-box access to the underlying distribution. Our framework unfolds in two steps. First, we develop algorithms for a restricted version of each problem, in which all possible scenarios are explicitly provided; second, we exploit structural properties of these algorithms and generalize them to the black-box setting. These key properties are: \textbf{(1)} the algorithms produce ``simple'' exponential families of black-box strategies, and \textbf{(2)} there exist \emph{efficient} ways to extend their output to the black-box case, which also preserve the approximation ratio exactly. We note that prior generalization approaches, i.e., variants of the \emph{Sample Average Approximation} method, can be used for the problems we consider, however they would yield worse approximation guarantees.