In this work we investigate the min-max-min robust optimization problem and the k-adaptability robust optimization problem for binary problems with uncertain costs. The idea of the first approach is to calculate a set of k feasible solutions which are worst-case optimal if in each possible scenario the best of the k solutions is implemented. It is known that the min-max-min robust problem can be solved efficiently if k is at least the dimension of the problem, while it is theoretically and computationally hard if k is small. However, nothing is known about the intermediate case, i.e. k lies between one and the dimension of the problem. We approach this open question and present an approximation algorithm which achieves good problem-specific approximation guarantees for the cases where k is close to or where k is a fraction of the dimension. The derived bounds can be used to show that the min-max-min robust problem is solvable in oracle-polynomial time under certain conditions even if k is smaller than the dimension. We extend the previous results to the robust k-adaptability problem. As a consequence we can provide bounds on the number of necessary second-stage policies to approximate the exact two-stage robust problem. We derive an approximation algorithm for the k-adaptability problem which has similar guarantees as for the min-max-min problem. Finally, we test both algorithms on knapsack and shortest path problems and related two-stage variants. The experiments show that both algorithms calculate solutions with relatively small optimality gap in seconds.

翻译：本文研究了成本不确定的二元问题中的最小-最大-最小鲁棒优化问题及K-适应性鲁棒优化问题。第一种方法的思想是计算一组k个可行解，这些解在每种可能情景下执行最优解时具有最坏情况最优性。已知当k至少等于问题维度时，最小-最大-最小鲁棒问题可高效求解，而k较小时在理论和计算上均具有难度。然而，对于中间情形（即k介于1与问题维度之间）尚无认知。我们针对这一开放问题提出近似算法，在k接近维度或为维度分数时，该算法可获得良好的问题特定近似保证。推导的边界可用于证明：即使k小于维度，在某些条件下最小-最大-最小鲁棒问题仍可在神谕多项式时间内求解。我们将前期结果推广至鲁棒K-适应性问题，从而能够给出逼近精确两阶段鲁棒问题所需第二阶段策略数量的边界。针对K-适应性问题导出的近似算法具有与最小-最大-最小问题相似的保证。最后，我们在背包问题、最短路径问题及其相关两阶段变体上测试了两种算法。实验表明，两种算法均可在数秒内计算出最优性间隙较小的解。