We present a nested local search algorithm to approximate several variants of metric two-stage stochastic facility location problems. These problems are generalizations of the well-studied metric uncapacitated facility location problem, taking uncertainties in demand values and costs into account. The proposed nested local search procedure uses three facility operations: adding, dropping, and swapping. To the best of our knowledge, this is the first constant-factor local search approximation for two-stage stochastic facility location problems. Besides traditional direct assignments from clients to facilities, we also investigate shared connections via capacitated trees and tours. We obtain the first constant-factor approximation algorithms for both connection types in the setting of two-stage stochastic optimization. Our algorithms admit order-preserving metrics and thus significantly generalize and improve the allowed mutability of the metric in comparison to previous algorithms, which only allow scenario-dependent inflation factors.
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