An Approximation Scheme for Stochastic Linear Programming and its Application to Stochastic Integer Programs

Algorithms for Optimization with Incomplete Information

计算机科学;物理学

David B. Shmoys ; Chaitanya Swamy

Others  :  http://drops.dagstuhl.de/opus/volltexte/2005/72/pdf/05031.SwamyChaitanya.Paper.72.pdf
PID  :  6821

学科分类：计算机科学（综合）

来源: CEUR

【 摘 要 】

Stochastic optimization problems attempt to model uncertainty in the data by assuming that the input is specified by a probability distribution. We consider the well studied paradigm of 2-stage models with recourse: first, given only distributional information about (some of) the data one commits on initial actions, and then once the actual data is realized (according to the distribution), further (recourse) actions can be taken. We show that for a broad class of 2-stage linear models with recourse, one can, for any, in time polynomial inand the size of the input, compute a solution of value within a factor of the optimum, in spite of the fact that exponentially many second-stage scenarios may occur. In conjunction with a suitable rounding scheme, this yields the first approximation algorithms for 2-stage stochastic integer optimization problems where the underlying random data is given by a "black box" and no restrictions are placed on the costs in the two stages. Our rounding approach for stochastic integer programs shows that an approximation algorithm for a deterministic analogue yields, with a small constant-factor loss, provably near-optimal solutions for the stochastic generalization. Among the range of applications we consider are stochastic versions of the multicommodity flow, set covering, and facility location problems.