On the complexity of a class of combinatorial optimization problems with uncertainty

We consider a robust (minmax-regret) version of the problem of selecting p elements of minimum total weight out of a set of In elements with uncertainty in weights of the elements. We present a polynomial algorithm with the order of complexity O((min {p, m - p})(2) m) for the case where uncertainty is represented by means of interval estimates for the weights. We show that the problem is NP-hard in the case of an arbitrary finite set of possible scenarios, even if there are only two possible scenarios. This is the first known example of a robust combinatorial optimization problem that is NP-hard in the case of scenario-represented uncertainty bur is polynomially solvable in the case of the interval representation of uncertainty.