Approximation Guarantees for Min-Max-Min Robust Optimization and \({\boldsymbol{k}}\)-Adaptability Under Objective Uncertainty
研究了二元问题在成本不确定下的最小-最大-最小鲁棒优化和k-适应性鲁棒优化，提出了针对中间k值的近似算法，并在背包和最短路径问题上验证了效果。
In this work we investigate the min-max-min robust optimization problem and the <i>k</i>-adaptability robust optimization problem for binary problems with uncertain costs. The idea of the first approach is to calculate a set of <i>k</i> feasible solutions which are worst-case optimal if in each possible scenario the best of the <i>k</i> solutions is implemented. It is known that the min-max-min robust problem can be solved efficiently if <i>k</i> is at least the dimension of the problem, while it is theoretically and computationally hard if <i>k</i> is small. However, nothing is known about the intermediate case, i.e., <i>k</i> 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 <i>k</i> is close to or 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 <i>k</i> is smaller than the dimension. We extend the previous results to the robust <i>k</i>-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 <i>k</i>-adaptability problem which has similar guarantees as for the min-max-min problem. Finally, we test both algorithms on knapsack and shortest path problems. The experiments show that both algorithms calculate solutions with relatively small optimality gap in seconds.