Approximation in Stochastic Integer Programming

Approximation algorithms are the prevalent solution methods in the field of stochastic programming. Problems in this field are very hard to solve. Indeed, most of the research in this field has concentrated on designing solution methods that approximate the optimal solutions. However, efficiency in the complexity theoretical sense is usually not taken into account. Quality statements mostly remain restricted to convergence to an optimal solution without accompanying implications on the running time of the algorithms for attaining more and more accurate solutions. However, over the last twenty years also some studies on performance analysis of approximation algorithms for stochastic programming have appeared. In this direction we find both probabilistic analysis and worst-case analysis. There have been studies on performance ratios and on absolute divergence from optimality. Recently the complexity of stochastic programming problems has been addressed, indeed confirming that these problems are harder than most combinatorial optimization problems. Polynomial time approximation algorithms and their performance guarantees for stochastic linear and integer programming problems have seen increasing research attention only very recently. Approximation in the traditional stochastic programming sense will not be discussed in this chapter. The reader interested in this issue is referred to surveys on stochastic programming, like the Handbook on Stochastic Programming [38] or the text books [3, 21, 35]. We concentrate on the studies of approximation algorithms which are more similar in nature to those for combinatorial optimization.