Conclusions and Open Problems

In this part of the monograph we have discussed a particular class of discrete optimization problems in which every feasible solution can be expressed as a subset of a given generic set E and the criterion, in the deterministic case, is the sum of the weights. The first important property of the minmax regret version of this problem is that the maximal regret of every feasible solution can be computed in polynomial time if the underlying problem \(\mathcal{P}\) is polynomially solvable. It results from a simple characterization of the worst case scenario of a given solution. The second important property of this class of problems is the preprocessing based on the concepts of possibly and necessary optimal elements. It turns out that the number of nonpossibly optimal elements may be quite large and removing them may significantly speed up calculations. Unfortunately, detecting all nonpossibly optimal elements may be itself NP-hard. For some particular problems, however, like Minmax Regret Minimum Spanning Tree or Minmax Regret Minimum Selecting Items, all nonpossibly and necessarily optimal elements can be efficiently detected.