Approximation in two-stage 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 solution value. 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 thirty 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. Recently the complexity of stochastic programming problems has been addressed, indeed confirming that these problems are harder than most deterministic combinatorial optimization problems. Polynomial-time approximation algorithms and their performance guarantees for stochastic linear and integer programming problems have received 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 by Ruszczyński and Shapiro (2003) or the textbooks by Birge and Louveaux (1997), Kall and Wallace (1994), Prékopa (1995), and Shapiro et al. (2009). We concentrate on the studies of approximation algorithms in relation to computational complexity theory. With this survey we intend to give a flavor of the type of results existing in the literature on approximation algorithms in two-stage stochastic integer programming rather than a complete overview of the literature on the subject. We do so by exhibiting a representative selection of results, which we present in full detail. While presenting them we do not refer to the literature; these references, together with pointers to other relevant work in this field of research, are given in an extensive notes section at the end of the survey