Robust combinatorial optimization with knapsack uncertainty

We study in this paper min max robust combinatorial optimization problems for an uncertainty polytope that is defined by knapsack constraints, either in the space of the optimization variables or in an extended space. We provide exact and approximation algorithms that extend the iterative algorithms proposed by Bertsimas and Sim (2003). We also study the limitation of the approach and point out NP-hard situations. Then, we approximate axis-parallel ellipsoids with knapsack constraints and provide an approximation scheme for the corresponding robust problem. The approximation scheme is also adapted to handle the intersection of an axis-parallel ellipsoid and a box.