Constructing Representative Scenarios to Approximate Robust Combinatorial Optimization Problems

In robust combinatorial optimization with discrete uncertainty, two general approximation algorithms are frequently used, which are both based on constructing a single scenario representing the whole uncertainty set. In the midpoint method, one optimizes for the average case scenario. In the element-wise worst-case approach, one constructs a scenario by taking the worst case in each component over all scenarios. Both methods are known to be $N$-approximations, where $N$ is the number of scenarios. In this paper, these results are refined by reconsidering their respective proofs as optimization problems. We present a linear program to construct a representative scenario for the uncertainty set, which guarantees an approximation guarantee that is at least as good as for the previous methods. Incidentally, we show that the element-wise worst-case approach can have an advantage over the midpoint approach if the number of scenarios is large. In numerical experiments on the selection problem we demonstrate that our approach can improve the approximation guarantee of the midpoint approach by around 20%.