Approximation Algorithms for 2-stage and Multi-stage Stochastic Optimization

Stochastic optimization problems provide a means to model uncertainty in the input data where the uncertainty is modeled by a probability distribution over the possible realizations of the data. We consider the well-studied paradigm of stochastic recourse models, in which the realized input is revealed through a series of stages and one can take decisions in each stage in response to the new information learned. We obtain the first approximation algorithms for a variety of 2-stage and k-stage stochastic linear and integer optimization problems where the underlying random data is given by a “black box” and no restrictions are placed on the recourse costs: one can merely sample data from this distribution, but no direct information about the distributions is given. Our contributions are twofold. First, we give a fully polynomial approximation scheme for solving a broad class of 2-stage and k-stage linear programs, where k is not part of the input, that is, we show that using only sampling access to the underlying distribution, one can, for any > 0, compute a solution of cost guaranteed to be within a (1+ ) factor of the optimum, in time polynomial in 1 and the size of the input. To the best of our knowledge, this is the first such result that shows that (a class) of multi-stage stochastic programs can be solved to near-optimality in polynomial time. Second, we give a rounding approach for stochastic integer programs that shows that approximation algorithms for a deterministic analogue yields, with a small constant-factor loss, provably near-optimal solutions for the stochastic generalization. Thus we obtain approximation algorithms for several stochastic problems, including the stochastic versions of the set cover, vertex cover, facility location, multicut (on trees) and multicommodity flow problems.