Approximation Algorithms for Reliable Stochastic Combinatorial Optimization

Abstract

We consider optimization problems that can be formulated as minimizing the cost of a feasible solution w ^T x over an arbitrary combinatorial feasible set ${ \mathcal{F} } \subset\{0, 1\}^n$. For these problems we describe a broad class of corresponding stochastic problems where the cost vector W has independent random components, unknown at the time of solution. A natural and important objective that incorporates risk in this stochastic setting is to look for a feasible solution whose stochastic cost has a small tail or a small convex combination of mean and standard deviation. Our models can be equivalently reformulated as nonconvex programs for which no efficient algorithms are known. In this paper, we make progress on these hard problems.

Our results are several efficient general-purpose approximation schemes. They use as a black-box (exact or approximate) the solution to the underlying deterministic problem and thus immediately apply to arbitrary combinatorial problems. For example, from an available δ-approximation algorithm to the linear problem, we construct a δ(1 + ε)-approximation algorithm for the stochastic problem, which invokes the linear algorithm only a logarithmic number of times in the problem input (and polynomial in $\frac{1}{\epsilon}$), for any desired accuracy level ε> 0. The algorithms are based on a geometric analysis of the curvature and approximability of the nonlinear level sets of the objective functions.