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In robust combinatorial optimization with discrete uncertainty, two generalapproximation algorithms are frequently used, which are both based onconstructing a single scenario representing the whole uncertainty set. In themidpoint method, one optimizes for the average case scenario. In theelement-wise worst-case approach, one constructs a scenario by taking the worstcase 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 respectiveproofs as optimization problems. We present a linear program to construct arepresentative scenario for the uncertainty set, which guarantees anapproximation guarantee that is at least as good as for the previous methods.Incidentally, we show that the element-wise worst-case approach can have anadvantage over the midpoint approach if the number of scenarios is large. Innumerical experiments on the selection problem we demonstrate that our approachcan improve the approximation guarantee of the midpoint approach by around 20%.

Constructing Representative Scenarios to Approximate Robust Combinatorial Optimization Problems