Approximation Algorithms for 2-Stage Stochastic Scheduling Problems

Abstract

There has been a series of results deriving approximation algorithms for 2-stage discrete stochastic optimization problems, in which the probabilistic component of the input is given by means of “black box”, from which the algorithm “learns” the distribution by drawing (a polynomial number of ) independent samples. The performance guarantees proved for such problems, of course, is generally worse than for their deterministic analogue. We focus on a 2-stage stochastic generalization of the problem of finding the maximum-weight subset of jobs that can be scheduled on one machine where each job is constrained to be processed within a specified time window. Surprisingly, we show that for this generalization, the same performance guarantee that is obtained for the deterministic case can be obtained for its stochastic extension.

Our algorithm builds on an approach of Charikar, Chekuri, and Pál: one first designs an approximation algorithm for the so-called polynomial scenario model (in which the probability distribution is restricted to have the property that there are only a polynomial number of possible realizations of the input that occur with positive probability); then one shows that by sampling from the distribution via the “black box” to obtain an approximate distribution that falls in this class and approximately solves this approximation to the problem, one nonetheless obtains a near-optimal solution to the original problem. Of course, to follow this broad outline, one must design an approximation algorithm for the stochastic optimization problem in the polynomial scenario model, and we do this by extending a result of Bar-Noy, Bar-Yehuda, Freund, Naor, and Schieber.

Furthermore, the results of Bar-Noy et al. extend to a wide variety of resource-constrained selection problems including, for example, the unrelated parallel-machine generalization R|r _j | ∑ w _j U _j and point-to-point admission control routing in networks (but with a different performance guarantee). Our techniques can also be extended to yield analogous results for the 2-stage stochastic generalizations for this class of problems.