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.

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