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We consider robust combinatorial optimization problems with cost uncertaintywhere the decision maker can prepare K solutions beforehand and chooses thebest of them once the true cost is revealed. Also known as min-max-minrobustness (a special case of K-adaptability), it is a viable alternative tootherwise intractable two-stage problems. The uncertainty set assumed in thispaper considers that in any scenario, at most Gamma of the components of thecost vectors will be higher than expected, which corresponds to the extremepoints of the budgeted uncertainty set.  While the classical min-max problem with budgeted uncertainty is essentiallyas easy as the underlying deterministic problem, it turns out that themin-max-min problem is NPhard for many easy combinatorial optimizationproblems, and not approximable in general. We thus present an integerprogramming formulation for solving the problem through a row-and-columngeneration algorithm. While exact, this algorithm can only cope with smallproblems, so we present two additional heuristics leveraging the structure ofbudgeted uncertainty. We compare our row-and-column generation algorithm andour heuristics on knapsack and shortest path instances previously used in thescientific literature and find that the heuristics obtain good qualitysolutions in short computational times.

Min-Max-Min Robustness for Combinatorial Problems with Discrete Budgeted Uncertainty