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On Scenario Aggregation to Approximate Robust Optimization Problems

As most robust combinatorial min-max and min-max regret problems with discrete uncertainty sets are NP-hard, research into approximation algorithm and approximability bounds has been a fruitful area of recent work. A simple and well-known approximation algorithm is the midpoint method, where one takes the average over all scenarios, and solves a problem of nominal type. Despite its simplicity, this method still gives the best-known bound on a wide range of problems, such as robust shortest path, or robust assignment problems. In this paper we present a simple extension of the midpoint method based on scenario aggregation, which improves the current best $K$-approximation result to an $(\varepsilon K)$-approximation for any desired $\varepsilon > 0$. Our method can be applied to min-max as well as min-max regret problems.