On the complexity of min–max–min robustness with two alternatives and budgeted uncertainty

Abstract

We study robust solutions for combinatorial optimization problems with budgeted uncertainty sets in the min–max–min setting, where the decision maker is allowed to choose a set of k different solutions from which one can be chosen after the uncertain scenario has been revealed.We first show how the problem can be decomposed into polynomially many subproblems if k is fixed. We then consider the special case where k=2, i.e., one is allowed to choose two different solutions to hedge against the uncertainty. Using the decomposition, we provide polynomial-time algorithms for the unconstrained combinatorial optimization problem, the matroid maximization problem, the selection problem, and the shortest path problem on series–parallel graphs. The shortest path problem on general graphs turns out to be NP-hard.We show how to transform approximation algorithms for minimization subproblems to approximation algorithms for the robust problem and study the knapsack problem to show that this does not hold for maximization problems in general. We present a PTAS for the corresponding subproblem and prove that the min–max–min knapsack problem with k=2 is not approximable at all.