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The main focus of this article is radius-based (supplier) clustering in the two-stage stochastic setting with recourse, where the inherent stochasticity of the model comes in the form of a budget constraint. In addition to the standard (homogeneous) setting where all clients must be within a distance\(R\)of the nearest facility, we provide results for the more general problem where the radius demands may beinhomogeneous(i.e., different for each client). We also explore a number of variants where additional constraints are imposed on the first-stage decisions, specifically matroid and multi-knapsack constraints, and provide results for these settings. We derive results for the most general distributional setting, where there is only black-box access to the underlying distribution. To accomplish this, we first develop algorithms for thepolynomial scenariossetting; we then employ a novelscenario-discardingvariant of the standardSample Average Approximationmethod, which crucially exploits properties of the restricted-case algorithms. We note that the scenario-discarding modification to the SAA method is necessary to optimize over the radius. 

In response to COVID-19, many countries have mandated social distancing and banned large group gatherings in order to slow down the spread of SARS-CoV-2. These social interventions along with vaccines remain the best way forward to reduce the spread of SARS CoV-2. In order to increase vaccine accessibility, states such as Virginia have deployed mobile vaccination centers to distribute vaccines across the state. When choosing where to place these sites, there are two important factors to take into account: accessibility and equity. We formulate a combinatorial problem that captures these factors and then develop efficient algorithms with theoretical guarantees on both of these aspects. Furthermore, we study the inherent hardness of the problem, and demonstrate strong impossibility results. Finally, we run computational experiments on real-world data to show the efficacy of our methods. 

Metric clustering is fundamental in areas ranging from Combinatorial Optimization and Data Mining, to Machine Learning and Operations Research. However, in a variety of situations we may have additional requirements or knowledge, distinct from the underlying metric, regarding which pairs of points should be clustered together. To capture and analyze such scenarios, we introduce a novel family of stochastic pairwise constraints, which we incorporate into several essential clustering objectives (radius/median/means). Moreover, we demonstrate that these constraints can succinctly model an intriguing collection of applications, including among others Individual Fairness in clustering and Must-link constraints in semi-supervised learning. Our main result consists of a general framework that yields approximation algorithms with provable guarantees for important clustering objectives, while at the same time producing solutions that respect the stochastic pairwise constraints. Furthermore, for certain objectives we devise improved results in the case of Must-link constraints, which are also the best possible from a theoretical perspective. Finally, we present experimental evidence that validates the effectiveness of our algorithms.