We consider various stochastic models that incorporate the notion of risk-averseness into the standard 2-stage recourse model, and develop novel techniques for solving the algorithmic problems arising in these models. A key notable feature of our work that distinguishes it from work in some other related models, such as the (standard) budget model and the (demand-) robust model, is that we obtain results in the black-box setting, that is, where one is given only sampling access to the underlying distribution. Our first model, which we call the risk-averse budget model, incorporates the notion of risk-averseness via a probabilistic constraint that restricts the probability (according to the underlying distribution) with which the second-stage cost may exceed a given budget B to at most a given input threshold ρ. We also a consider a closely-related model that we call the risk-averse robust model, where we seek to minimize the first-stage cost and the (1 − ρ)-quantile (according to the distribution) of the second-stage cost. We obtain approximation algorithms for a variety of combinatorial optimization problems including the set cover, vertex cover, multicut on trees, min cut, and facility location problems, in the risk-averse budget and robust models with black-box distributions. We first devise a fully polynomial approximation scheme for solving the LP-relaxations of a wide-variety of risk-averse budgeted problems. Complementing this, we give a rounding procedure that lets us use existing LP-based approximation algorithms for the 2-stage stochastic and/or deterministic counterpart of the problem to round the fractional solution. Thus, we obtain near-optimal solutions to risk-averse problems that preserve the budget approximately and incur a small blow-up of the probability threshold (both of which are unavoidable). To the best of our knowledge, these are the first approximation results for problems involving probabilistic constraints and black-box distributions. Our results extend to the setting with non-uniform scenario-budgets, and to a generalization of the risk-averse robust model, where the goal is to minimize the sum of the first-stage cost and a weighted combination of the expectation and the (1 − ρ)-quantile of the second-stage cost.
Specifically, in contrast to the results in REF , we do not incur any approximation in the probabilistic constraint, and the running time of our algorithms is independent of the input threshold ρ.
Algorithms for Probabilistically-Constrained Models of Risk-Averse Stochastic Optimization with Black-Box Distributions
ArXiv
ArXiv
Mathematics Computer Science