New complexity results and algorithms for min-max-min robust combinatorial optimization
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In this work we investigate the min-max-min robust optimization problem applied to combinatorial problems with uncertain cost-vectors which are contained in a convex uncertainty set. The idea of the 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 would be 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. While both cases are well studied in the literature nothing is known about the intermediate case, namely if k is smaller than but close to the dimension of the problem. We approach this open question and show that for a selection of combinatorial problems the min-max-min problem can be solved exactly and approximately in polynomial time if some problem specific values are fixed. Furthermore we approach a second open question and present the first implementable algorithm with oracle-pseudopolynomial runtime for the case that k is at least the dimension of the problem. The algorithm is based on a projected subgradient method where the projection problem is solved by the classical Frank-Wolfe algorithm. Additionally we derive a branch bound method to solve the min-max-min problem for arbitrary values of k and perform tests on knapsack and shortest path instances. The experiments show that despite its theoretical impact the projected subgradient method cannot compete with an already existing method. On the other hand the performance of the branch bound method scales very well with the number of solutions. Thus we are able to solve instances where k is above some small threshold very efficiently.
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