We propose a model for recoverable robust optimization with commitment. Given a combinatorial optimization problem and uncertainty about elements that may fail, we ask for a robust solution that, after the failing elements are revealed, can be augmented in a limited way. However, we commit to preserve the remaining elements of the initial solution. We consider different polynomial-time solvable combinatorial optimization problems and settle the computational complexity of their robust counterparts with commitment. We show for the weighted matroid basis problem that an optimal solution to the nominal problem is also optimal for its robust counterpart. Indeed, matroids are provably the only structures with this strong property. Robust counterparts of other problems are NP-hard such as the matching and the stable set problem, even in bipartite graphs. However, we establish polynomial-time algorithms for the robust counterparts of the unweighted stable set problem in bipartite graphs and the weighted stable set problem in interval graphs, also known as the interval scheduling problem.

翻译：我们提出了一种带承诺的可恢复鲁棒优化模型。给定一个组合优化问题以及可能失效的不确定元素，我们要求得到一个鲁棒解，该解在失效元素被揭示后，能够以有限的方式进行扩充。然而，我们承诺保留初始解中的其余元素。我们考虑了不同的多项式时间可解的组合优化问题，并确定了其带承诺的鲁棒对应问题的计算复杂性。我们证明，对于加权拟阵基问题，名义问题的最优解对其鲁棒对应问题也是最优的。事实上，拟阵被证明是唯一具有此强性质的结构。其他问题的鲁棒对应问题是NP难的，例如匹配问题和稳定集问题，即使在二分图中也是如此。然而，我们为二分图中无权稳定集问题和区间图中的加权稳定集问题（也称为区间调度问题）的鲁棒对应问题建立了多项式时间算法。