Fully Polynomial Time Approximation Schemes for Robust Multistage Decision Making

We design a framework to obtain Fully Polynomial Time Approximation Schemes (FPTASes) for adjustable robust multistage decision making under the budgeted uncertainty sets introduced by Bertsimas and Sim. We apply this framework to the robust counterpart of three problems coming from operations research: (i) ordered knapsack, (ii) single-item inventory control, and (iii) single-item batch dispatch. Our work gives the first FPTAS for these problems, and for adjustable robust multistage decision making in general. The proposed approximation schemes are constructed with the technique of K-approximation sets and functions, relying on careful robust dynamic programming formulations for a master problem (corresponding to the decision maker) and for an adversary problem (corresponding to nature, which chooses bad realizations of uncertainty for the decision maker). The resulting algorithms are short and simple, requiring just a few concise subroutines.